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Theorem List for Metamath Proof Explorer - 39301-39400   *Has distinct variable group(s)
TypeLabelDescription
Statement

Theoremee23an 39301 e23an 39300 without virtual deductions. (Contributed by Alan Sare, 14-Jul-2011.) (Proof modification is discouraged.) (New usage is discouraged.)
(𝜑 → (𝜓𝜒))    &   (𝜑 → (𝜓 → (𝜃𝜏)))    &   ((𝜒𝜏) → 𝜂)       (𝜑 → (𝜓 → (𝜃𝜂)))

Theoreme32 39302 A virtual deduction elimination rule. (Contributed by Alan Sare, 12-Jun-2011.) (Proof modification is discouraged.) (New usage is discouraged.)
(   𝜑   ,   𝜓   ,   𝜒   ▶   𝜃   )    &   (   𝜑   ,   𝜓   ▶   𝜏   )    &   (𝜃 → (𝜏𝜂))       (   𝜑   ,   𝜓   ,   𝜒   ▶   𝜂   )

Theoremee32 39303 e32 39302 without virtual deductions. (Contributed by Alan Sare, 18-Jul-2011.) (Proof modification is discouraged.) (New usage is discouraged.)
(𝜑 → (𝜓 → (𝜒𝜃)))    &   (𝜑 → (𝜓𝜏))    &   (𝜃 → (𝜏𝜂))       (𝜑 → (𝜓 → (𝜒𝜂)))

Theoreme32an 39304 A virtual deduction elimination rule. (Contributed by Alan Sare, 24-Jun-2011.) (Proof modification is discouraged.) (New usage is discouraged.)
(   𝜑   ,   𝜓   ,   𝜒   ▶   𝜃   )    &   (   𝜑   ,   𝜓   ▶   𝜏   )    &   ((𝜃𝜏) → 𝜂)       (   𝜑   ,   𝜓   ,   𝜒   ▶   𝜂   )

Theoremee32an 39305 e33an 39279 without virtual deductions. (Contributed by Alan Sare, 14-Jul-2011.) (Proof modification is discouraged.) (New usage is discouraged.)
(𝜑 → (𝜓 → (𝜒𝜃)))    &   (𝜑 → (𝜓𝜏))    &   ((𝜃𝜏) → 𝜂)       (𝜑 → (𝜓 → (𝜒𝜂)))

Theoreme123 39306 A virtual deduction elimination rule. (Contributed by Alan Sare, 12-Jun-2011.) (Proof modification is discouraged.) (New usage is discouraged.)
(   𝜑   ▶   𝜓   )    &   (   𝜑   ,   𝜒   ▶   𝜃   )    &   (   𝜑   ,   𝜒   ,   𝜏   ▶   𝜂   )    &   (𝜓 → (𝜃 → (𝜂𝜁)))       (   𝜑   ,   𝜒   ,   𝜏   ▶   𝜁   )

Theoremee123 39307 e123 39306 without virtual deductions. (Contributed by Alan Sare, 25-Jul-2011.) (Proof modification is discouraged.) (New usage is discouraged.)
(𝜑𝜓)    &   (𝜑 → (𝜒𝜃))    &   (𝜑 → (𝜒 → (𝜏𝜂)))    &   (𝜓 → (𝜃 → (𝜂𝜁)))       (𝜑 → (𝜒 → (𝜏𝜁)))

Theoremel123 39308 A virtual deduction elimination rule. (Contributed by Alan Sare, 13-Jun-2015.) (Proof modification is discouraged.) (New usage is discouraged.)
(   𝜑   ▶   𝜓   )    &   (   𝜒   ▶   𝜃   )    &   (   𝜏   ▶   𝜂   )    &   ((𝜓𝜃𝜂) → 𝜁)       (   (   𝜑   ,   𝜒   ,   𝜏   )   ▶   𝜁   )

Theoreme233 39309 A virtual deduction elimination rule. (Contributed by Alan Sare, 29-Feb-2012.) (Proof modification is discouraged.) (New usage is discouraged.)
(   𝜑   ,   𝜓   ▶   𝜒   )    &   (   𝜑   ,   𝜓   ,   𝜃   ▶   𝜏   )    &   (   𝜑   ,   𝜓   ,   𝜃   ▶   𝜂   )    &   (𝜒 → (𝜏 → (𝜂𝜁)))       (   𝜑   ,   𝜓   ,   𝜃   ▶   𝜁   )

Theoreme323 39310 A virtual deduction elimination rule. (Contributed by Alan Sare, 17-Apr-2012.) (Proof modification is discouraged.) (New usage is discouraged.)
(   𝜑   ,   𝜓   ,   𝜒   ▶   𝜃   )    &   (   𝜑   ,   𝜓   ▶   𝜏   )    &   (   𝜑   ,   𝜓   ,   𝜒   ▶   𝜂   )    &   (𝜃 → (𝜏 → (𝜂𝜁)))       (   𝜑   ,   𝜓   ,   𝜒   ▶   𝜁   )

Theoreme000 39311 A virtual deduction elimination rule. The non-virtual deduction form of e000 39311 is the virtual deduction form. (Contributed by Alan Sare, 14-Jun-2011.) (Proof modification is discouraged.) (New usage is discouraged.)
𝜑    &   𝜓    &   𝜒    &   (𝜑 → (𝜓 → (𝜒𝜃)))       𝜃

Theoreme00 39312 Elimination rule identical to mp2 9. The non-virtual deduction form is the virtual deduction form, which is mp2 9. (Contributed by Alan Sare, 14-Jun-2011.) (Proof modification is discouraged.) (New usage is discouraged.)
𝜑    &   𝜓    &   (𝜑 → (𝜓𝜒))       𝜒

Theoreme00an 39313 Elimination rule identical to mp2an 708. The non-virtual deduction form is the virtual deduction form, which is mp2an 708. (Contributed by Alan Sare, 15-Jun-2011.) (Proof modification is discouraged.) (New usage is discouraged.)
𝜑    &   𝜓    &   ((𝜑𝜓) → 𝜒)       𝜒

Theoremeel00cT 39314 An elimination deduction. (Contributed by Alan Sare, 4-Feb-2017.) (Proof modification is discouraged.) (New usage is discouraged.)
𝜑    &   𝜓    &   ((𝜑𝜓) → 𝜒)       (⊤ → 𝜒)

TheoremeelTT 39315 An elimination deduction. (Contributed by Alan Sare, 4-Feb-2017.) (Proof modification is discouraged.) (New usage is discouraged.)
(⊤ → 𝜑)    &   (⊤ → 𝜓)    &   ((𝜑𝜓) → 𝜒)       𝜒

Theoreme0a 39316 Elimination rule identical to ax-mp 5. The non-virtual deduction form is the virtual deduction form, which is ax-mp 5. (Contributed by Alan Sare, 14-Jun-2011.) (Proof modification is discouraged.) (New usage is discouraged.)
𝜑    &   (𝜑𝜓)       𝜓

TheoremeelT 39317 An elimination deduction. (Contributed by Alan Sare, 5-Feb-2017.) (Proof modification is discouraged.) (New usage is discouraged.)
(⊤ → 𝜑)    &   (𝜑𝜓)       𝜓

Theoremeel0cT 39318 An elimination deduction. (Contributed by Alan Sare, 4-Feb-2017.) (Proof modification is discouraged.) (New usage is discouraged.)
𝜑    &   (𝜑𝜓)       (⊤ → 𝜓)

TheoremeelT0 39319 An elimination deduction. (Contributed by Alan Sare, 4-Feb-2017.) (Proof modification is discouraged.) (New usage is discouraged.)
(⊤ → 𝜑)    &   𝜓    &   ((𝜑𝜓) → 𝜒)       𝜒

Theoreme0bi 39320 Elimination rule identical to mpbi 220. The non-virtual deduction form is the virtual deduction form, which is mpbi 220. (Contributed by Alan Sare, 15-Jun-2011.) (Proof modification is discouraged.) (New usage is discouraged.)
𝜑    &   (𝜑𝜓)       𝜓

Theoreme0bir 39321 Elimination rule identical to mpbir 221. The non-virtual deduction form is the virtual deduction form, which is mpbir 221. (Contributed by Alan Sare, 15-Jun-2011.) (Proof modification is discouraged.) (New usage is discouraged.)
𝜑    &   (𝜓𝜑)       𝜓

Theoremuun0.1 39322 Convention notation form of un0.1 39323. (Contributed by Alan Sare, 23-Apr-2015.) (Proof modification is discouraged.) (New usage is discouraged.)
(⊤ → 𝜑)    &   (𝜓𝜒)    &   ((⊤ ∧ 𝜓) → 𝜃)       (𝜓𝜃)

Theoremun0.1 39323 is the constant true, a tautology (see df-tru 1526). Kleene's "empty conjunction" is logically equivalent to . In a virtual deduction we shall interpret to be the empty wff or the empty collection of virtual hypotheses. in a virtual deduction translated into conventional notation we shall interpret to be Kleene's empty conjunction. If 𝜃 is true given the empty collection of virtual hypotheses and another collection of virtual hypotheses, then it is true given only the other collection of virtual hypotheses. (Contributed by Alan Sare, 23-Apr-2015.) (Proof modification is discouraged.) (New usage is discouraged.)
(      ▶   𝜑   )    &   (   𝜓   ▶   𝜒   )    &   (   (      ,   𝜓   )   ▶   𝜃   )       (   𝜓   ▶   𝜃   )

TheoremuunT1 39324 A deduction unionizing a non-unionized collection of virtual hypotheses. (Contributed by Alan Sare, 3-Dec-2015.) Proof was revised to accommodate a possible future version of df-tru 1526. (Revised by David A. Wheeler, 8-May-2019.) (Proof modification is discouraged.) (New usage is discouraged.)
((⊤ ∧ 𝜑) → 𝜓)       (𝜑𝜓)

TheoremuunT1p1 39325 A deduction unionizing a non-unionized collection of virtual hypotheses. (Contributed by Alan Sare, 4-Feb-2017.) (Proof modification is discouraged.) (New usage is discouraged.)
((𝜑 ∧ ⊤) → 𝜓)       (𝜑𝜓)

TheoremuunT21 39326 A deduction unionizing a non-unionized collection of virtual hypotheses. (Contributed by Alan Sare, 3-Dec-2015.) (Proof modification is discouraged.) (New usage is discouraged.) (Proof modification is discouraged.) (New usage is discouraged.)
((⊤ ∧ (𝜑𝜓)) → 𝜒)       ((𝜑𝜓) → 𝜒)

Theoremuun121 39327 A deduction unionizing a non-unionized collection of virtual hypotheses. (Contributed by Alan Sare, 4-Feb-2017.) (Proof modification is discouraged.) (New usage is discouraged.)
((𝜑 ∧ (𝜑𝜓)) → 𝜒)       ((𝜑𝜓) → 𝜒)

Theoremuun121p1 39328 A deduction unionizing a non-unionized collection of virtual hypotheses. (Contributed by Alan Sare, 4-Feb-2017.) (Proof modification is discouraged.) (New usage is discouraged.)
(((𝜑𝜓) ∧ 𝜑) → 𝜒)       ((𝜑𝜓) → 𝜒)

Theoremuun132 39329 A deduction unionizing a non-unionized collection of virtual hypotheses. (Contributed by Alan Sare, 4-Feb-2017.) (Proof modification is discouraged.) (New usage is discouraged.)
((𝜑 ∧ (𝜓𝜒)) → 𝜃)       ((𝜑𝜓𝜒) → 𝜃)

Theoremuun132p1 39330 A deduction unionizing a non-unionized collection of virtual hypotheses. (Contributed by Alan Sare, 4-Feb-2017.) (Proof modification is discouraged.) (New usage is discouraged.)
(((𝜓𝜒) ∧ 𝜑) → 𝜃)       ((𝜑𝜓𝜒) → 𝜃)

Theoremanabss7p1 39331 A deduction unionizing a non-unionized collection of virtual hypotheses. This would have been named uun221 if the 0th permutation did not exist in set.mm as anabss7 879. (Contributed by Alan Sare, 4-Feb-2017.) (Proof modification is discouraged.) (New usage is discouraged.)
(((𝜓𝜑) ∧ 𝜑) → 𝜒)       ((𝜓𝜑) → 𝜒)

Theoremun10 39332 A unionizing deduction. (Contributed by Alan Sare, 28-Apr-2015.) (Proof modification is discouraged.) (New usage is discouraged.)
(   (   𝜑   ,      )   ▶   𝜓   )       (   𝜑   ▶   𝜓   )

Theoremun01 39333 A unionizing deduction. (Contributed by Alan Sare, 28-Apr-2015.) (Proof modification is discouraged.) (New usage is discouraged.)
(   (      ,   𝜑   )   ▶   𝜓   )       (   𝜑   ▶   𝜓   )

Theoremun2122 39334 A deduction unionizing a non-unionized collection of virtual hypotheses. (Contributed by Alan Sare, 3-Dec-2015.) (Proof modification is discouraged.) (New usage is discouraged.)
(((𝜑𝜓) ∧ 𝜓𝜓) → 𝜒)       ((𝜑𝜓) → 𝜒)

Theoremuun2131 39335 A deduction unionizing a non-unionized collection of virtual hypotheses. (Contributed by Alan Sare, 4-Feb-2017.) (Proof modification is discouraged.) (New usage is discouraged.)
(((𝜑𝜓) ∧ (𝜑𝜒)) → 𝜃)       ((𝜑𝜓𝜒) → 𝜃)

Theoremuun2131p1 39336 A deduction unionizing a non-unionized collection of virtual hypotheses. (Contributed by Alan Sare, 4-Feb-2017.) (Proof modification is discouraged.) (New usage is discouraged.)
(((𝜑𝜒) ∧ (𝜑𝜓)) → 𝜃)       ((𝜑𝜓𝜒) → 𝜃)

TheoremuunTT1 39337 A deduction unionizing a non-unionized collection of virtual hypotheses. (Contributed by Alan Sare, 4-Feb-2017.) (Proof modification is discouraged.) (New usage is discouraged.)
((⊤ ∧ ⊤ ∧ 𝜑) → 𝜓)       (𝜑𝜓)

TheoremuunTT1p1 39338 A deduction unionizing a non-unionized collection of virtual hypotheses. (Contributed by Alan Sare, 4-Feb-2017.) (Proof modification is discouraged.) (New usage is discouraged.)
((⊤ ∧ 𝜑 ∧ ⊤) → 𝜓)       (𝜑𝜓)

TheoremuunTT1p2 39339 A deduction unionizing a non-unionized collection of virtual hypotheses. (Contributed by Alan Sare, 4-Feb-2017.) (Proof modification is discouraged.) (New usage is discouraged.)
((𝜑 ∧ ⊤ ∧ ⊤) → 𝜓)       (𝜑𝜓)

TheoremuunT11 39340 A deduction unionizing a non-unionized collection of virtual hypotheses. (Contributed by Alan Sare, 4-Feb-2017.) (Proof modification is discouraged.) (New usage is discouraged.)
((⊤ ∧ 𝜑𝜑) → 𝜓)       (𝜑𝜓)

TheoremuunT11p1 39341 A deduction unionizing a non-unionized collection of virtual hypotheses. (Contributed by Alan Sare, 4-Feb-2017.) (Proof modification is discouraged.) (New usage is discouraged.)
((𝜑 ∧ ⊤ ∧ 𝜑) → 𝜓)       (𝜑𝜓)

TheoremuunT11p2 39342 A deduction unionizing a non-unionized collection of virtual hypotheses. (Contributed by Alan Sare, 4-Feb-2017.) (Proof modification is discouraged.) (New usage is discouraged.)
((𝜑𝜑 ∧ ⊤) → 𝜓)       (𝜑𝜓)

TheoremuunT12 39343 A deduction unionizing a non-unionized collection of virtual hypotheses. (Contributed by Alan Sare, 4-Feb-2017.) (Proof modification is discouraged.) (New usage is discouraged.)
((⊤ ∧ 𝜑𝜓) → 𝜒)       ((𝜑𝜓) → 𝜒)

TheoremuunT12p1 39344 A deduction unionizing a non-unionized collection of virtual hypotheses. (Contributed by Alan Sare, 4-Feb-2017.) (Proof modification is discouraged.) (New usage is discouraged.)
((⊤ ∧ 𝜓𝜑) → 𝜒)       ((𝜑𝜓) → 𝜒)

TheoremuunT12p2 39345 A deduction unionizing a non-unionized collection of virtual hypotheses. (Contributed by Alan Sare, 4-Feb-2017.) (Proof modification is discouraged.) (New usage is discouraged.)
((𝜑 ∧ ⊤ ∧ 𝜓) → 𝜒)       ((𝜑𝜓) → 𝜒)

TheoremuunT12p3 39346 A deduction unionizing a non-unionized collection of virtual hypotheses. (Contributed by Alan Sare, 4-Feb-2017.) (Proof modification is discouraged.) (New usage is discouraged.)
((𝜓 ∧ ⊤ ∧ 𝜑) → 𝜒)       ((𝜑𝜓) → 𝜒)

TheoremuunT12p4 39347 A deduction unionizing a non-unionized collection of virtual hypotheses. (Contributed by Alan Sare, 4-Feb-2017.) (Proof modification is discouraged.) (New usage is discouraged.)
((𝜑𝜓 ∧ ⊤) → 𝜒)       ((𝜑𝜓) → 𝜒)

TheoremuunT12p5 39348 A deduction unionizing a non-unionized collection of virtual hypotheses. (Contributed by Alan Sare, 4-Feb-2017.) (Proof modification is discouraged.) (New usage is discouraged.)
((𝜓𝜑 ∧ ⊤) → 𝜒)       ((𝜑𝜓) → 𝜒)

Theoremuun111 39349 A deduction unionizing a non-unionized collection of virtual hypotheses. (Contributed by Alan Sare, 4-Feb-2017.) (Proof modification is discouraged.) (New usage is discouraged.)
((𝜑𝜑𝜑) → 𝜓)       (𝜑𝜓)

Theorem3anidm12p1 39350 A deduction unionizing a non-unionized collection of virtual hypotheses. 3anidm12 1423 denotes the deduction which would have been named uun112 if it did not pre-exist in set.mm. This second permutation's name is based on this pre-existing name. (Contributed by Alan Sare, 4-Feb-2017.) (Proof modification is discouraged.) (New usage is discouraged.)
((𝜑𝜓𝜑) → 𝜒)       ((𝜑𝜓) → 𝜒)

Theorem3anidm12p2 39351 A deduction unionizing a non-unionized collection of virtual hypotheses. (Contributed by Alan Sare, 4-Feb-2017.) (Proof modification is discouraged.) (New usage is discouraged.)
((𝜓𝜑𝜑) → 𝜒)       ((𝜑𝜓) → 𝜒)

Theoremuun123 39352 A deduction unionizing a non-unionized collection of virtual hypotheses. (Contributed by Alan Sare, 4-Feb-2017.) (Proof modification is discouraged.) (New usage is discouraged.)
((𝜑𝜒𝜓) → 𝜃)       ((𝜑𝜓𝜒) → 𝜃)

Theoremuun123p1 39353 A deduction unionizing a non-unionized collection of virtual hypotheses. (Contributed by Alan Sare, 4-Feb-2017.) (Proof modification is discouraged.) (New usage is discouraged.)
((𝜓𝜑𝜒) → 𝜃)       ((𝜑𝜓𝜒) → 𝜃)

Theoremuun123p2 39354 A deduction unionizing a non-unionized collection of virtual hypotheses. (Contributed by Alan Sare, 4-Feb-2017.) (Proof modification is discouraged.) (New usage is discouraged.)
((𝜒𝜑𝜓) → 𝜃)       ((𝜑𝜓𝜒) → 𝜃)

Theoremuun123p3 39355 A deduction unionizing a non-unionized collection of virtual hypotheses. (Contributed by Alan Sare, 4-Feb-2017.) (Proof modification is discouraged.) (New usage is discouraged.)
((𝜓𝜒𝜑) → 𝜃)       ((𝜑𝜓𝜒) → 𝜃)

Theoremuun123p4 39356 A deduction unionizing a non-unionized collection of virtual hypotheses. (Contributed by Alan Sare, 4-Feb-2017.) (Proof modification is discouraged.) (New usage is discouraged.)
((𝜒𝜓𝜑) → 𝜃)       ((𝜑𝜓𝜒) → 𝜃)

Theoremuun2221 39357 A deduction unionizing a non-unionized collection of virtual hypotheses. (Contributed by Alan Sare, 30-Dec-2016.) (Proof modification is discouraged.) (New usage is discouraged.) (Proof modification is discouraged.) (New usage is discouraged.)
((𝜑𝜑 ∧ (𝜓𝜑)) → 𝜒)       ((𝜓𝜑) → 𝜒)

Theoremuun2221p1 39358 A deduction unionizing a non-unionized collection of virtual hypotheses. (Contributed by Alan Sare, 4-Feb-2017.) (Proof modification is discouraged.) (New usage is discouraged.)
((𝜑 ∧ (𝜓𝜑) ∧ 𝜑) → 𝜒)       ((𝜓𝜑) → 𝜒)

Theoremuun2221p2 39359 A deduction unionizing a non-unionized collection of virtual hypotheses. (Contributed by Alan Sare, 4-Feb-2017.) (Proof modification is discouraged.) (New usage is discouraged.)
(((𝜓𝜑) ∧ 𝜑𝜑) → 𝜒)       ((𝜓𝜑) → 𝜒)

Theorem3impdirp1 39360 A deduction unionizing a non-unionized collection of virtual hypotheses. Commuted version of 3impdir 1422. (Contributed by Alan Sare, 4-Feb-2017.) (Proof modification is discouraged.) (New usage is discouraged.)
(((𝜒𝜓) ∧ (𝜑𝜓)) → 𝜃)       ((𝜑𝜒𝜓) → 𝜃)

Theorem3impcombi 39361 A 1-hypothesis propositional calculus deduction. (Contributed by Alan Sare, 25-Sep-2017.)
((𝜑𝜓𝜑) → (𝜒𝜃))       ((𝜓𝜑𝜒) → 𝜃)

20.31.6  Theorems proved using Virtual Deduction

TheoremtrsspwALT 39362 Virtual deduction proof of the left-to-right implication of dftr4 4790. A transitive class is a subset of its power class. This proof corresponds to the virtual deduction proof of dftr4 4790 without accumulating results. (Contributed by Alan Sare, 29-Apr-2011.) (Proof modification is discouraged.) (New usage is discouraged.)
(Tr 𝐴𝐴 ⊆ 𝒫 𝐴)

TheoremtrsspwALT2 39363 Virtual deduction proof of trsspwALT 39362. This proof is the same as the proof of trsspwALT 39362 except each virtual deduction symbol is replaced by its non-virtual deduction symbol equivalent. A transitive class is a subset of its power class. (Contributed by Alan Sare, 23-Jul-2011.) (Proof modification is discouraged.) (New usage is discouraged.)
(Tr 𝐴𝐴 ⊆ 𝒫 𝐴)

TheoremtrsspwALT3 39364 Short predicate calculus proof of the left-to-right implication of dftr4 4790. A transitive class is a subset of its power class. This proof was constructed by applying Metamath's minimize command to the proof of trsspwALT2 39363, which is the virtual deduction proof trsspwALT 39362 without virtual deductions. (Contributed by Alan Sare, 30-Apr-2011.) (Proof modification is discouraged.) (New usage is discouraged.)
(Tr 𝐴𝐴 ⊆ 𝒫 𝐴)

Theoremsspwtr 39365 Virtual deduction proof of the right-to-left implication of dftr4 4790. A class which is a subclass of its power class is transitive. This proof corresponds to the virtual deduction proof of sspwtr 39365 without accumulating results. (Contributed by Alan Sare, 2-May-2011.) (Proof modification is discouraged.) (New usage is discouraged.)
(𝐴 ⊆ 𝒫 𝐴 → Tr 𝐴)

TheoremsspwtrALT 39366 Virtual deduction proof of sspwtr 39365. This proof is the same as the proof of sspwtr 39365 except each virtual deduction symbol is replaced by its non-virtual deduction symbol equivalent. A class which is a subclass of its power class is transitive. (Contributed by Alan Sare, 3-May-2011.) (Proof modification is discouraged.) (New usage is discouraged.)
(𝐴 ⊆ 𝒫 𝐴 → Tr 𝐴)

TheoremcsbabgOLD 39367* Move substitution into a class abstraction. (Contributed by NM, 13-Dec-2005.) (Proof shortened by Andrew Salmon, 9-Jul-2011.) Obsolete as of 19-Aug-2018. Use csbab 4041 instead. (New usage is discouraged.) (Proof modification is discouraged.)
(𝐴𝑉𝐴 / 𝑥{𝑦𝜑} = {𝑦[𝐴 / 𝑥]𝜑})

TheoremcsbxpgOLD 39368 Distribute proper substitution through the Cartesian product of two classes. (Contributed by Alan Sare, 10-Nov-2012.) Obsolete as of 23-Aug-2018. Use csbrn 5631 instead. (New usage is discouraged.) (Proof modification is discouraged.)
(𝐴𝐷𝐴 / 𝑥(𝐵 × 𝐶) = (𝐴 / 𝑥𝐵 × 𝐴 / 𝑥𝐶))

TheoremcsbingOLD 39369 Distribute proper substitution through an intersection relation. (Contributed by Alan Sare, 22-Jul-2012.) Obsolete as of 18-Aug-2018. Use csbin 4043 instead. (New usage is discouraged.) (Proof modification is discouraged.)
(𝐴𝐵𝐴 / 𝑥(𝐶𝐷) = (𝐴 / 𝑥𝐶𝐴 / 𝑥𝐷))

TheoremcsbresgOLD 39370 Distribute proper substitution through the restriction of a class. csbresgOLD 39370 is derived from the virtual deduction proof csbresgVD 39445. (Contributed by Alan Sare, 10-Nov-2012.) Obsolete as of 23-Aug-2018. Use csbres 5431 instead. (New usage is discouraged.) (Proof modification is discouraged.)
(𝐴𝑉𝐴 / 𝑥(𝐵𝐶) = (𝐴 / 𝑥𝐵𝐴 / 𝑥𝐶))

TheoremcsbrngOLD 39371 Distribute proper substitution through the range of a class. (Contributed by Alan Sare, 10-Nov-2012.) Obsolete as of 23-Aug-2018. Use csbrn 5631 instead. (New usage is discouraged.) (Proof modification is discouraged.)
(𝐴𝑉𝐴 / 𝑥ran 𝐵 = ran 𝐴 / 𝑥𝐵)

TheoremsspwtrALT2 39372 Short predicate calculus proof of the right-to-left implication of dftr4 4790. A class which is a subclass of its power class is transitive. This proof was constructed by applying Metamath's minimize command to the proof of sspwtrALT 39366, which is the virtual deduction proof sspwtr 39365 without virtual deductions. (Contributed by Alan Sare, 3-May-2011.) (Proof modification is discouraged.) (New usage is discouraged.)
(𝐴 ⊆ 𝒫 𝐴 → Tr 𝐴)

TheorempwtrVD 39373 Virtual deduction proof of pwtr 4951; see pwtrrVD 39374 for the converse. (Contributed by Alan Sare, 25-Aug-2011.) (Proof modification is discouraged.) (New usage is discouraged.)
(Tr 𝐴 → Tr 𝒫 𝐴)

TheorempwtrrVD 39374 Virtual deduction proof of pwtr 4951; see pwtrVD 39373 for the converse. (Contributed by Alan Sare, 25-Aug-2011.) (Proof modification is discouraged.) (New usage is discouraged.)
𝐴 ∈ V       (Tr 𝒫 𝐴 → Tr 𝐴)

TheoremsuctrALT 39375 The successor of a transitive class is transitive. The proof of http://us.metamath.org/other/completeusersproof/suctrvd.html is a Virtual Deduction proof verified by automatically transforming it into the Metamath proof of suctrALT 39375 using completeusersproof, which is verified by the Metamath program. The proof of http://us.metamath.org/other/completeusersproof/suctrro.html is a form of the completed proof which preserves the Virtual Deduction proof's step numbers and their ordering. See suctr 5846 for the original proof. (Contributed by Alan Sare, 11-Apr-2009.) (Revised by Alan Sare, 12-Jun-2018.) (Proof modification is discouraged.) (New usage is discouraged.)
(Tr 𝐴 → Tr suc 𝐴)

TheoremsnssiALTVD 39376 Virtual deduction proof of snssiALT 39377. (Contributed by Alan Sare, 11-Sep-2011.) (Proof modification is discouraged.) (New usage is discouraged.)
(𝐴𝐵 → {𝐴} ⊆ 𝐵)

TheoremsnssiALT 39377 If a class is an element of another class, then its singleton is a subclass of that other class. Alternate proof of snssi 4371. This theorem was automatically generated from snssiALTVD 39376 using a translation program. (Contributed by Alan Sare, 11-Sep-2011.) (Proof modification is discouraged.) (New usage is discouraged.)
(𝐴𝐵 → {𝐴} ⊆ 𝐵)

TheoremsnsslVD 39378 Virtual deduction proof of snssl 39379. (Contributed by Alan Sare, 25-Aug-2011.) (Proof modification is discouraged.) (New usage is discouraged.)
𝐴 ∈ V       ({𝐴} ⊆ 𝐵𝐴𝐵)

Theoremsnssl 39379 If a singleton is a subclass of another class, then the singleton's element is an element of that other class. This theorem is the right-to-left implication of the biconditional snss 4348. The proof of this theorem was automatically generated from snsslVD 39378 using a tools command file, translateMWO.cmd, by translating the proof into its non-virtual deduction form and minimizing it. (Contributed by Alan Sare, 25-Aug-2011.) (Proof modification is discouraged.) (New usage is discouraged.)
𝐴 ∈ V       ({𝐴} ⊆ 𝐵𝐴𝐵)

TheoremsnelpwrVD 39380 Virtual deduction proof of snelpwi 4942. (Contributed by Alan Sare, 25-Aug-2011.) (Proof modification is discouraged.) (New usage is discouraged.)
(𝐴𝐵 → {𝐴} ∈ 𝒫 𝐵)

TheoremunipwrVD 39381 Virtual deduction proof of unipwr 39382. (Contributed by Alan Sare, 25-Aug-2011.) (Proof modification is discouraged.) (New usage is discouraged.)
𝐴 𝒫 𝐴

Theoremunipwr 39382 A class is a subclass of the union of its power class. This theorem is the right-to-left subclass lemma of unipw 4948. The proof of this theorem was automatically generated from unipwrVD 39381 using a tools command file , translateMWO.cmd , by translating the proof into its non-virtual deduction form and minimizing it. (Contributed by Alan Sare, 25-Aug-2011.) (Proof modification is discouraged.) (New usage is discouraged.)
𝐴 𝒫 𝐴

TheoremsstrALT2VD 39383 Virtual deduction proof of sstrALT2 39384. (Contributed by Alan Sare, 11-Sep-2011.) (Proof modification is discouraged.) (New usage is discouraged.)
((𝐴𝐵𝐵𝐶) → 𝐴𝐶)

TheoremsstrALT2 39384 Virtual deduction proof of sstr 3644, transitivity of subclasses, Theorem 6 of [Suppes] p. 23. This theorem was automatically generated from sstrALT2VD 39383 using the command file translatewithout_overwriting.cmd . It was not minimized because the automated minimization excluding duplicates generates a minimized proof which, although not directly containing any duplicates, indirectly contains a duplicate. That is, the trace back of the minimized proof contains a duplicate. This is undesirable because some step(s) of the minimized proof use the proven theorem. (Contributed by Alan Sare, 11-Sep-2011.) (Proof modification is discouraged.) (New usage is discouraged.)
((𝐴𝐵𝐵𝐶) → 𝐴𝐶)

TheoremsuctrALT2VD 39385 Virtual deduction proof of suctrALT2 39386. (Contributed by Alan Sare, 11-Sep-2011.) (Proof modification is discouraged.) (New usage is discouraged.)
(Tr 𝐴 → Tr suc 𝐴)

TheoremsuctrALT2 39386 Virtual deduction proof of suctr 5846. The sucessor of a transitive class is transitive. This proof was generated automatically from the virtual deduction proof suctrALT2VD 39385 using the tools command file translatewithout_overwritingminimize_excludingduplicates.cmd . (Contributed by Alan Sare, 11-Sep-2011.) (Proof modification is discouraged.) (New usage is discouraged.)
(Tr 𝐴 → Tr suc 𝐴)

Theoremelex2VD 39387* Virtual deduction proof of elex2 3247. (Contributed by Alan Sare, 25-Sep-2011.) (Proof modification is discouraged.) (New usage is discouraged.)
(𝐴𝐵 → ∃𝑥 𝑥𝐵)

Theoremelex22VD 39388* Virtual deduction proof of elex22 3248. (Contributed by Alan Sare, 24-Oct-2011.) (Proof modification is discouraged.) (New usage is discouraged.)
((𝐴𝐵𝐴𝐶) → ∃𝑥(𝑥𝐵𝑥𝐶))

Theoremeqsbc3rVD 39389* Virtual deduction proof of eqsbc3r 3525. (Contributed by Alan Sare, 24-Oct-2011.) (Proof modification is discouraged.) (New usage is discouraged.)
(𝐴𝐵 → ([𝐴 / 𝑥]𝐶 = 𝑥𝐶 = 𝐴))

Theoremzfregs2VD 39390* Virtual deduction proof of zfregs2 8647. (Contributed by Alan Sare, 24-Oct-2011.) (Proof modification is discouraged.) (New usage is discouraged.)
(𝐴 ≠ ∅ → ¬ ∀𝑥𝐴𝑦(𝑦𝐴𝑦𝑥))

Theoremtpid3gVD 39391 Virtual deduction proof of tpid3g 4337. (Contributed by Alan Sare, 24-Oct-2011.) (Proof modification is discouraged.) (New usage is discouraged.)
(𝐴𝐵𝐴 ∈ {𝐶, 𝐷, 𝐴})

Theoremen3lplem1VD 39392* Virtual deduction proof of en3lplem1 8549. (Contributed by Alan Sare, 24-Oct-2011.) (Proof modification is discouraged.) (New usage is discouraged.)
((𝐴𝐵𝐵𝐶𝐶𝐴) → (𝑥 = 𝐴 → ∃𝑦(𝑦 ∈ {𝐴, 𝐵, 𝐶} ∧ 𝑦𝑥)))

Theoremen3lplem2VD 39393* Virtual deduction proof of en3lplem2 8550. (Contributed by Alan Sare, 24-Oct-2011.) (Proof modification is discouraged.) (New usage is discouraged.)
((𝐴𝐵𝐵𝐶𝐶𝐴) → (𝑥 ∈ {𝐴, 𝐵, 𝐶} → ∃𝑦(𝑦 ∈ {𝐴, 𝐵, 𝐶} ∧ 𝑦𝑥)))

Theoremen3lpVD 39394 Virtual deduction proof of en3lp 8551. (Contributed by Alan Sare, 24-Oct-2011.) (Proof modification is discouraged.) (New usage is discouraged.)
¬ (𝐴𝐵𝐵𝐶𝐶𝐴)

20.31.7  Theorems proved using Virtual Deduction with mmj2 assistance

Theoremsimplbi2VD 39395 Virtual deduction proof of simplbi2 654. The following user's proof is completed by invoking mmj2's unify command and using mmj2's StepSelector to pick all remaining steps of the Metamath proof.
 h1:: ⊢ (𝜑 ↔ (𝜓 ∧ 𝜒)) 3:1,?: e0a 39316 ⊢ ((𝜓 ∧ 𝜒) → 𝜑) qed:3,?: e0a 39316 ⊢ (𝜓 → (𝜒 → 𝜑))
The proof of simplbi2 654 was automatically derived from it. (Contributed by Alan Sare, 31-Dec-2011.) (Proof modification is discouraged.) (New usage is discouraged.)
(𝜑 ↔ (𝜓𝜒))       (𝜓 → (𝜒𝜑))

Theorem3ornot23VD 39396 Virtual deduction proof of 3ornot23 39032. The following user's proof is completed by invoking mmj2's unify command and using mmj2's StepSelector to pick all remaining steps of the Metamath proof.
1::
 ⊢ (   (¬ 𝜑 ∧ ¬ 𝜓)   ▶   (¬ 𝜑 ∧ ¬ 𝜓)   ) 2:: ⊢ (   (¬ 𝜑 ∧ ¬ 𝜓)   ,   (𝜒 ∨ 𝜑 ∨ 𝜓)   ▶   (𝜒 ∨ 𝜑 ∨ 𝜓)   ) 3:1,?: e1a 39169 ⊢ (   (¬ 𝜑 ∧ ¬ 𝜓)   ▶   ¬ 𝜑   ) 4:1,?: e1a 39169 ⊢ (   (¬ 𝜑 ∧ ¬ 𝜓)   ▶   ¬ 𝜓   ) 5:3,4,?: e11 39230 ⊢ (   (¬ 𝜑 ∧ ¬ 𝜓)   ▶   ¬ (𝜑 ∨ 𝜓)   ) 6:2,?: e2 39173 ⊢ (   (¬ 𝜑 ∧ ¬ 𝜓)   ,   (𝜒 ∨ 𝜑 ∨ 𝜓)   ▶   (𝜒 ∨ (𝜑 ∨ 𝜓))   ) 7:5,6,?: e12 39268 ⊢ (   (¬ 𝜑 ∧ ¬ 𝜓)   ,   (𝜒 ∨ 𝜑 ∨ 𝜓)   ▶   𝜒   ) 8:7: ⊢ (   (¬ 𝜑 ∧ ¬ 𝜓)   ▶   ((𝜒 ∨ 𝜑 ∨ 𝜓) → 𝜒)   ) qed:8: ⊢ ((¬ 𝜑 ∧ ¬ 𝜓) → ((𝜒 ∨ 𝜑 ∨ 𝜓) → 𝜒))
(Contributed by Alan Sare, 31-Dec-2011.) (Proof modification is discouraged.) (New usage is discouraged.)
((¬ 𝜑 ∧ ¬ 𝜓) → ((𝜒𝜑𝜓) → 𝜒))

Theoremorbi1rVD 39397 Virtual deduction proof of orbi1r 39033. The following user's proof is completed by invoking mmj2's unify command and using mmj2's StepSelector to pick all remaining steps of the Metamath proof.
 1:: ⊢ (   (𝜑 ↔ 𝜓)   ▶   (𝜑 ↔ 𝜓)   ) 2:: ⊢ (   (𝜑 ↔ 𝜓)   ,   (𝜒 ∨ 𝜑)    ▶   (𝜒 ∨ 𝜑)   ) 3:2,?: e2 39173 ⊢ (   (𝜑 ↔ 𝜓)   ,   (𝜒 ∨ 𝜑)    ▶   (𝜑 ∨ 𝜒)   ) 4:1,3,?: e12 39268 ⊢ (   (𝜑 ↔ 𝜓)   ,   (𝜒 ∨ 𝜑)    ▶   (𝜓 ∨ 𝜒)   ) 5:4,?: e2 39173 ⊢ (   (𝜑 ↔ 𝜓)   ,   (𝜒 ∨ 𝜑)    ▶   (𝜒 ∨ 𝜓)   ) 6:5: ⊢ (   (𝜑 ↔ 𝜓)   ▶   ((𝜒 ∨ 𝜑) → (𝜒 ∨ 𝜓))   ) 7:: ⊢ (   (𝜑 ↔ 𝜓)   ,   (𝜒 ∨ 𝜓)    ▶   (𝜒 ∨ 𝜓)   ) 8:7,?: e2 39173 ⊢ (   (𝜑 ↔ 𝜓)   ,   (𝜒 ∨ 𝜓)    ▶   (𝜓 ∨ 𝜒)   ) 9:1,8,?: e12 39268 ⊢ (   (𝜑 ↔ 𝜓)   ,   (𝜒 ∨ 𝜓)    ▶   (𝜑 ∨ 𝜒)   ) 10:9,?: e2 39173 ⊢ (   (𝜑 ↔ 𝜓)   ,   (𝜒 ∨ 𝜓)    ▶   (𝜒 ∨ 𝜑)   ) 11:10: ⊢ (   (𝜑 ↔ 𝜓)   ▶   ((𝜒 ∨ 𝜓) → (𝜒 ∨ 𝜑))   ) 12:6,11,?: e11 39230 ⊢ (   (𝜑 ↔ 𝜓)   ▶   ((𝜒 ∨ 𝜑) ↔ (𝜒 ∨ 𝜓))   ) qed:12: ⊢ ((𝜑 ↔ 𝜓) → ((𝜒 ∨ 𝜑) ↔ (𝜒 ∨ 𝜓)))
(Contributed by Alan Sare, 31-Dec-2011.) (Proof modification is discouraged.) (New usage is discouraged.)
((𝜑𝜓) → ((𝜒𝜑) ↔ (𝜒𝜓)))

Theorembitr3VD 39398 Virtual deduction proof of bitr3 341. The following user's proof is completed by invoking mmj2's unify command and using mmj2's StepSelector to pick all remaining steps of the Metamath proof.
 1:: ⊢ (   (𝜑 ↔ 𝜓)   ▶   (𝜑 ↔ 𝜓)   ) 2:1,?: e1a 39169 ⊢ (   (𝜑 ↔ 𝜓)   ▶   (𝜓 ↔ 𝜑)   ) 3:: ⊢ (   (𝜑 ↔ 𝜓)   ,   (𝜑 ↔ 𝜒)    ▶   (𝜑 ↔ 𝜒)   ) 4:3,?: e2 39173 ⊢ (   (𝜑 ↔ 𝜓)   ,   (𝜑 ↔ 𝜒)    ▶   (𝜒 ↔ 𝜑)   ) 5:2,4,?: e12 39268 ⊢ (   (𝜑 ↔ 𝜓)   ,   (𝜑 ↔ 𝜒)    ▶   (𝜓 ↔ 𝜒)   ) 6:5: ⊢ (   (𝜑 ↔ 𝜓)   ▶   ((𝜑 ↔ 𝜒) → (𝜓 ↔ 𝜒))   ) qed:6: ⊢ ((𝜑 ↔ 𝜓) → ((𝜑 ↔ 𝜒) → (𝜓 ↔ 𝜒)))
(Contributed by Alan Sare, 31-Dec-2011.) (Proof modification is discouraged.) (New usage is discouraged.)
((𝜑𝜓) → ((𝜑𝜒) → (𝜓𝜒)))

Theorem3orbi123VD 39399 Virtual deduction proof of 3orbi123 39034. The following user's proof is completed by invoking mmj2's unify command and using mmj2's StepSelector to pick all remaining steps of the Metamath proof.
 1:: ⊢ (   ((𝜑 ↔ 𝜓) ∧ (𝜒 ↔ 𝜃) ∧ (𝜏 ↔ 𝜂))   ▶   ((𝜑 ↔ 𝜓) ∧ (𝜒 ↔ 𝜃) ∧ (𝜏 ↔ 𝜂))   ) 2:1,?: e1a 39169 ⊢ (   ((𝜑 ↔ 𝜓) ∧ (𝜒 ↔ 𝜃) ∧ (𝜏 ↔ 𝜂))   ▶   (𝜑 ↔ 𝜓)   ) 3:1,?: e1a 39169 ⊢ (   ((𝜑 ↔ 𝜓) ∧ (𝜒 ↔ 𝜃) ∧ (𝜏 ↔ 𝜂))   ▶   (𝜒 ↔ 𝜃)   ) 4:1,?: e1a 39169 ⊢ (   ((𝜑 ↔ 𝜓) ∧ (𝜒 ↔ 𝜃) ∧ (𝜏 ↔ 𝜂))   ▶   (𝜏 ↔ 𝜂)   ) 5:2,3,?: e11 39230 ⊢ (   ((𝜑 ↔ 𝜓) ∧ (𝜒 ↔ 𝜃) ∧ (𝜏 ↔ 𝜂))   ▶   ((𝜑 ∨ 𝜒) ↔ (𝜓 ∨ 𝜃))   ) 6:5,4,?: e11 39230 ⊢ (   ((𝜑 ↔ 𝜓) ∧ (𝜒 ↔ 𝜃) ∧ (𝜏 ↔ 𝜂))   ▶   (((𝜑 ∨ 𝜒) ∨ 𝜏) ↔ ((𝜓 ∨ 𝜃) ∨ 𝜂))   ) 7:?: ⊢ (((𝜑 ∨ 𝜒) ∨ 𝜏) ↔ (𝜑 ∨ 𝜒 ∨ 𝜏)) 8:6,7,?: e10 39236 ⊢ (   ((𝜑 ↔ 𝜓) ∧ (𝜒 ↔ 𝜃) ∧ (𝜏 ↔ 𝜂))   ▶   ((𝜑 ∨ 𝜒 ∨ 𝜏) ↔ ((𝜓 ∨ 𝜃) ∨ 𝜂))   ) 9:?: ⊢ (((𝜓 ∨ 𝜃) ∨ 𝜂) ↔ (𝜓 ∨ 𝜃 ∨ 𝜂)) 10:8,9,?: e10 39236 ⊢ (   ((𝜑 ↔ 𝜓) ∧ (𝜒 ↔ 𝜃) ∧ (𝜏 ↔ 𝜂))   ▶   ((𝜑 ∨ 𝜒 ∨ 𝜏) ↔ (𝜓 ∨ 𝜃 ∨ 𝜂))   ) qed:10: ⊢ (((𝜑 ↔ 𝜓) ∧ (𝜒 ↔ 𝜃) ∧ (𝜏 ↔ 𝜂)) → ((𝜑 ∨ 𝜒 ∨ 𝜏) ↔ (𝜓 ∨ 𝜃 ∨ 𝜂)))
(Contributed by Alan Sare, 31-Dec-2011.) (Proof modification is discouraged.) (New usage is discouraged.)
(((𝜑𝜓) ∧ (𝜒𝜃) ∧ (𝜏𝜂)) → ((𝜑𝜒𝜏) ↔ (𝜓𝜃𝜂)))

Theoremsbc3orgVD 39400 Virtual deduction proof of sbc3orgOLD 39059. The following user's proof is completed by invoking mmj2's unify command and using mmj2's StepSelector to pick all remaining steps of the Metamath proof.
 1:: ⊢ (   𝐴 ∈ 𝐵   ▶   𝐴 ∈ 𝐵   ) 2:1,?: e1a 39169 ⊢ (   𝐴 ∈ 𝐵   ▶   ([𝐴 / 𝑥]((𝜑 ∨ 𝜓) ∨ 𝜒) ↔ ([𝐴 / 𝑥](𝜑 ∨ 𝜓) ∨ [𝐴 / 𝑥]𝜒))   ) 3:: ⊢ (((𝜑 ∨ 𝜓) ∨ 𝜒) ↔ (𝜑 ∨ 𝜓 ∨ 𝜒)) 32:3: ⊢ ∀𝑥(((𝜑 ∨ 𝜓) ∨ 𝜒) ↔ (𝜑 ∨ 𝜓 ∨ 𝜒)) 33:1,32,?: e10 39236 ⊢ (   𝐴 ∈ 𝐵   ▶   [𝐴 / 𝑥](((𝜑 ∨ 𝜓) ∨ 𝜒) ↔ (𝜑 ∨ 𝜓 ∨ 𝜒))   ) 4:1,33,?: e11 39230 ⊢ (   𝐴 ∈ 𝐵   ▶   ([𝐴 / 𝑥]((𝜑 ∨ 𝜓) ∨ 𝜒) ↔ [𝐴 / 𝑥](𝜑 ∨ 𝜓 ∨ 𝜒))   ) 5:2,4,?: e11 39230 ⊢ (   𝐴 ∈ 𝐵   ▶   ([𝐴 / 𝑥](𝜑 ∨ 𝜓 ∨ 𝜒) ↔ ([𝐴 / 𝑥](𝜑 ∨ 𝜓) ∨ [𝐴 / 𝑥]𝜒))   ) 6:1,?: e1a 39169 ⊢ (   𝐴 ∈ 𝐵   ▶   ([𝐴 / 𝑥](𝜑 ∨ 𝜓) ↔ ([𝐴 / 𝑥]𝜑 ∨ [𝐴 / 𝑥]𝜓))   ) 7:6,?: e1a 39169 ⊢ (   𝐴 ∈ 𝐵   ▶   (([𝐴 / 𝑥](𝜑 ∨ 𝜓) ∨ [𝐴 / 𝑥]𝜒) ↔ (([𝐴 / 𝑥]𝜑 ∨ [𝐴 / 𝑥]𝜓) ∨ [𝐴 / 𝑥]𝜒))   ) 8:5,7,?: e11 39230 ⊢ (   𝐴 ∈ 𝐵   ▶   ([𝐴 / 𝑥](𝜑 ∨ 𝜓 ∨ 𝜒) ↔ (([𝐴 / 𝑥]𝜑 ∨ [𝐴 / 𝑥]𝜓) ∨ [𝐴 / 𝑥]𝜒))   ) 9:?: ⊢ ((([𝐴 / 𝑥]𝜑 ∨ [𝐴 / 𝑥]𝜓) ∨ [𝐴 / 𝑥]𝜒) ↔ ([𝐴 / 𝑥]𝜑 ∨ [𝐴 / 𝑥]𝜓 ∨ [𝐴 / 𝑥]𝜒)) 10:8,9,?: e10 39236 ⊢ (   𝐴 ∈ 𝐵   ▶   ([𝐴 / 𝑥](𝜑 ∨ 𝜓 ∨ 𝜒) ↔ ([𝐴 / 𝑥]𝜑 ∨ [𝐴 / 𝑥]𝜓 ∨ [𝐴 / 𝑥]𝜒))   ) qed:10: ⊢ (𝐴 ∈ 𝐵 → ([𝐴 / 𝑥](𝜑 ∨ 𝜓 ∨ 𝜒) ↔ ([𝐴 / 𝑥]𝜑 ∨ [𝐴 / 𝑥]𝜓 ∨ [𝐴 / 𝑥]𝜒)))
(Contributed by Alan Sare, 31-Dec-2011.) (Proof modification is discouraged.) (New usage is discouraged.)
(𝐴𝐵 → ([𝐴 / 𝑥](𝜑𝜓𝜒) ↔ ([𝐴 / 𝑥]𝜑[𝐴 / 𝑥]𝜓[𝐴 / 𝑥]𝜒)))

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78 7701-7800 79 7801-7900 80 7901-8000 81 8001-8100 82 8101-8200 83 8201-8300 84 8301-8400 85 8401-8500 86 8501-8600 87 8601-8700 88 8701-8800 89 8801-8900 90 8901-9000 91 9001-9100 92 9101-9200 93 9201-9300 94 9301-9400 95 9401-9500 96 9501-9600 97 9601-9700 98 9701-9800 99 9801-9900 100 9901-10000 101 10001-10100 102 10101-10200 103 10201-10300 104 10301-10400 105 10401-10500 106 10501-10600 107 10601-10700 108 10701-10800 109 10801-10900 110 10901-11000 111 11001-11100 112 11101-11200 113 11201-11300 114 11301-11400 115 11401-11500 116 11501-11600 117 11601-11700 118 11701-11800 119 11801-11900 120 11901-12000 121 12001-12100 122 12101-12200 123 12201-12300 124 12301-12400 125 12401-12500 126 12501-12600 127 12601-12700 128 12701-12800 129 12801-12900 130 12901-13000 131 13001-13100 132 13101-13200 133 13201-13300 134 13301-13400 135 13401-13500 136 13501-13600 137 13601-13700 138 13701-13800 139 13801-13900 140 13901-14000 141 14001-14100 142 14101-14200 143 14201-14300 144 14301-14400 145 14401-14500 146 14501-14600 147 14601-14700 148 14701-14800 149 14801-14900 150 14901-15000 151 15001-15100 152 15101-15200 153 15201-15300 154 15301-15400 155 15401-15500 156 15501-15600 157 15601-15700 158 15701-15800 159 15801-15900 160 15901-16000 161 16001-16100 162 16101-16200 163 16201-16300 164 16301-16400 165 16401-16500 166 16501-16600 167 16601-16700 168 16701-16800 169 16801-16900 170 16901-17000 171 17001-17100 172 17101-17200 173 17201-17300 174 17301-17400 175 17401-17500 176 17501-17600 177 17601-17700 178 17701-17800 179 17801-17900 180 17901-18000 181 18001-18100 182 18101-18200 183 18201-18300 184 18301-18400 185 18401-18500 186 18501-18600 187 18601-18700 188 18701-18800 189 18801-18900 190 18901-19000 191 19001-19100 192 19101-19200 193 19201-19300 194 19301-19400 195 19401-19500 196 19501-19600 197 19601-19700 198 19701-19800 199 19801-19900 200 19901-20000 201 20001-20100 202 20101-20200 203 20201-20300 204 20301-20400 205 20401-20500 206 20501-20600 207 20601-20700 208 20701-20800 209 20801-20900 210 20901-21000 211 21001-21100 212 21101-21200 213 21201-21300 214 21301-21400 215 21401-21500 216 21501-21600 217 21601-21700 218 21701-21800 219 21801-21900 220 21901-22000 221 22001-22100 222 22101-22200 223 22201-22300 224 22301-22400 225 22401-22500 226 22501-22600 227 22601-22700 228 22701-22800 229 22801-22900 230 22901-23000 231 23001-23100 232 23101-23200 233 23201-23300 234 23301-23400 235 23401-23500 236 23501-23600 237 23601-23700 238 23701-23800 239 23801-23900 240 23901-24000 241 24001-24100 242 24101-24200 243 24201-24300 244 24301-24400 245 24401-24500 246 24501-24600 247 24601-24700 248 24701-24800 249 24801-24900 250 24901-25000 251 25001-25100 252 25101-25200 253 25201-25300 254 25301-25400 255 25401-25500 256 25501-25600 257 25601-25700 258 25701-25800 259 25801-25900 260 25901-26000 261 26001-26100 262 26101-26200 263 26201-26300 264 26301-26400 265 26401-26500 266 26501-26600 267 26601-26700 268 26701-26800 269 26801-26900 270 26901-27000 271 27001-27100 272 27101-27200 273 27201-27300 274 27301-27400 275 27401-27500 276 27501-27600 277 27601-27700 278 27701-27800 279 27801-27900 280 27901-28000 281 28001-28100 282 28101-28200 283 28201-28300 284 28301-28400 285 28401-28500 286 28501-28600 287 28601-28700 288 28701-28800 289 28801-28900 290 28901-29000 291 29001-29100 292 29101-29200 293 29201-29300 294 29301-29400 295 29401-29500 296 29501-29600 297 29601-29700 298 29701-29800 299 29801-29900 300 29901-30000 301 30001-30100 302 30101-30200 303 30201-30300 304 30301-30400 305 30401-30500 306 30501-30600 307 30601-30700 308 30701-30800 309 30801-30900 310 30901-31000 311 31001-31100 312 31101-31200 313 31201-31300 314 31301-31400 315 31401-31500 316 31501-31600 317 31601-31700 318 31701-31800 319 31801-31900 320 31901-32000 321 32001-32100 322 32101-32200 323 32201-32300 324 32301-32400 325 32401-32500 326 32501-32600 327 32601-32700 328 32701-32800 329 32801-32900 330 32901-33000 331 33001-33100 332 33101-33200 333 33201-33300 334 33301-33400 335 33401-33500 336 33501-33600 337 33601-33700 338 33701-33800 339 33801-33900 340 33901-34000 341 34001-34100 342 34101-34200 343 34201-34300 344 34301-34400 345 34401-34500 346 34501-34600 347 34601-34700 348 34701-34800 349 34801-34900 350 34901-35000 351 35001-35100 352 35101-35200 353 35201-35300 354 35301-35400 355 35401-35500 356 35501-35600 357 35601-35700 358 35701-35800 359 35801-35900 360 35901-36000 361 36001-36100 362 36101-36200 363 36201-36300 364 36301-36400 365 36401-36500 366 36501-36600 367 36601-36700 368 36701-36800 369 36801-36900 370 36901-37000 371 37001-37100 372 37101-37200 373 37201-37300 374 37301-37400 375 37401-37500 376 37501-37600 377 37601-37700 378 37701-37800 379 37801-37900 380 37901-38000 381 38001-38100 382 38101-38200 383 38201-38300 384 38301-38400 385 38401-38500 386 38501-38600 387 38601-38700 388 38701-38800 389 38801-38900 390 38901-39000 391 39001-39100 392 39101-39200 393 39201-39300 394 39301-39400 395 39401-39500 396 39501-39600 397 39601-39700 398 39701-39800 399 39801-39900 400 39901-40000 401 40001-40100 402 40101-40200 403 40201-40300 404 40301-40400 405 40401-40500 406 40501-40600 407 40601-40700 408 40701-40800 409 40801-40900 410 40901-41000 411 41001-41100 412 41101-41200 413 41201-41300 414 41301-41400 415 41401-41500 416 41501-41600 417 41601-41700 418 41701-41800 419 41801-41900 420 41901-42000 421 42001-42100 422 42101-42200 423 42201-42300 424 42301-42400 425 42401-42500 426 42501-42600 427 42601-42700 428 42701-42800 429 42801-42879
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