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Theorem List for Metamath Proof Explorer - 39001-39100   *Has distinct variable group(s)
TypeLabelDescription
Statement

Theoremexbir 39001 Exportation implication also converting the consequent from a biconditional to an implication. Derived automatically from exbirVD 39402. (Contributed by Alan Sare, 31-Dec-2011.)
(((𝜑𝜓) → (𝜒𝜃)) → (𝜑 → (𝜓 → (𝜃𝜒))))

Theorem3impexpbicom 39002 Version of 3impexp 1311 where in addition the consequent is commuted. (Contributed by Alan Sare, 31-Dec-2011.)
(((𝜑𝜓𝜒) → (𝜃𝜏)) ↔ (𝜑 → (𝜓 → (𝜒 → (𝜏𝜃)))))

Theorem3impexpbicomi 39003 Inference associated with 3impexpbicom 39002. Derived automatically from 3impexpbicomiVD 39407. (Contributed by Alan Sare, 31-Dec-2011.)
((𝜑𝜓𝜒) → (𝜃𝜏))       (𝜑 → (𝜓 → (𝜒 → (𝜏𝜃))))

20.31.2  Supplementary unification deductions

Theorembi1imp 39004 Importation inference similar to imp 444, except the outermost implication of the hypothesis is a biconditional. (Contributed by Alan Sare, 6-Nov-2017.)
(𝜑 ↔ (𝜓𝜒))       ((𝜑𝜓) → 𝜒)

Theorembi2imp 39005 Importation inference similar to imp 444, except both implications of the hypothesis are biconditionals. (Contributed by Alan Sare, 6-Nov-2017.)
(𝜑 ↔ (𝜓𝜒))       ((𝜑𝜓) → 𝜒)

Theorembi3impb 39006 Similar to 3impb 1279 with implication in hypothesis replaced by biconditional. (Contributed by Alan Sare, 6-Nov-2017.)
((𝜑 ∧ (𝜓𝜒)) ↔ 𝜃)       ((𝜑𝜓𝜒) → 𝜃)

Theorembi3impa 39007 Similar to 3impa 1278 with implication in hypothesis replaced by biconditional. (Contributed by Alan Sare, 6-Nov-2017.)
(((𝜑𝜓) ∧ 𝜒) ↔ 𝜃)       ((𝜑𝜓𝜒) → 𝜃)

Theorembi23impib 39008 3impib 1281 with the inner implication of the hypothesis a biconditional. (Contributed by Alan Sare, 6-Nov-2017.)
(𝜑 → ((𝜓𝜒) ↔ 𝜃))       ((𝜑𝜓𝜒) → 𝜃)

Theorembi13impib 39009 3impib 1281 with the outer implication of the hypothesis a biconditional. (Contributed by Alan Sare, 6-Nov-2017.)
(𝜑 ↔ ((𝜓𝜒) → 𝜃))       ((𝜑𝜓𝜒) → 𝜃)

Theorembi123impib 39010 3impib 1281 with the implications of the hypothesis biconditionals. (Contributed by Alan Sare, 6-Nov-2017.)
(𝜑 ↔ ((𝜓𝜒) ↔ 𝜃))       ((𝜑𝜓𝜒) → 𝜃)

Theorembi13impia 39011 3impia 1280 with the outer implication of the hypothesis a biconditional. (Contributed by Alan Sare, 6-Nov-2017.)
((𝜑𝜓) ↔ (𝜒𝜃))       ((𝜑𝜓𝜒) → 𝜃)

Theorembi123impia 39012 3impia 1280 with the implications of the hypothesis biconditionals. (Contributed by Alan Sare, 6-Nov-2017.)
((𝜑𝜓) ↔ (𝜒𝜃))       ((𝜑𝜓𝜒) → 𝜃)

Theorembi33imp12 39013 3imp 1275 with innermost implication of the hypothesis a biconditional. (Contributed by Alan Sare, 6-Nov-2017.)
(𝜑 → (𝜓 → (𝜒𝜃)))       ((𝜑𝜓𝜒) → 𝜃)

Theorembi23imp13 39014 3imp 1275 with middle implication of the hypothesis a biconditional. (Contributed by Alan Sare, 6-Nov-2017.)
(𝜑 → (𝜓 ↔ (𝜒𝜃)))       ((𝜑𝜓𝜒) → 𝜃)

Theorembi13imp23 39015 3imp 1275 with outermost implication of the hypothesis a biconditional. (Contributed by Alan Sare, 6-Nov-2017.)
(𝜑 ↔ (𝜓 → (𝜒𝜃)))       ((𝜑𝜓𝜒) → 𝜃)

Theorembi13imp2 39016 Similar to 3imp 1275 except the outermost and innermost implications are biconditionals. (Contributed by Alan Sare, 6-Nov-2017.)
(𝜑 ↔ (𝜓 → (𝜒𝜃)))       ((𝜑𝜓𝜒) → 𝜃)

Theorembi12imp3 39017 Similar to 3imp 1275 except all but innermost implication are biconditionals. (Contributed by Alan Sare, 6-Nov-2017.)
(𝜑 ↔ (𝜓 ↔ (𝜒𝜃)))       ((𝜑𝜓𝜒) → 𝜃)

Theorembi23imp1 39018 Similar to 3imp 1275 except all but outermost implication are biconditionals. (Contributed by Alan Sare, 6-Nov-2017.)
(𝜑 → (𝜓 ↔ (𝜒𝜃)))       ((𝜑𝜓𝜒) → 𝜃)

Theorembi123imp0 39019 Similar to 3imp 1275 except all implications are biconditionals. (Contributed by Alan Sare, 6-Nov-2017.)
(𝜑 ↔ (𝜓 ↔ (𝜒𝜃)))       ((𝜑𝜓𝜒) → 𝜃)

Theorem4animp1 39020 A single hypothesis unification deduction with an assertion which is an implication with a 4-right-nested conjunction antecedent. (Contributed by Alan Sare, 30-May-2018.)
((𝜑𝜓𝜒) → (𝜏𝜃))       ((((𝜑𝜓) ∧ 𝜒) ∧ 𝜃) → 𝜏)

Theorem4an31 39021 A rearrangement of conjuncts for a 4-right-nested conjunction. (Contributed by Alan Sare, 30-May-2018.)
((((𝜒𝜓) ∧ 𝜑) ∧ 𝜃) → 𝜏)       ((((𝜑𝜓) ∧ 𝜒) ∧ 𝜃) → 𝜏)

Theorem4an4132 39022 A rearrangement of conjuncts for a 4-right-nested conjunction. (Contributed by Alan Sare, 30-May-2018.)
((((𝜃𝜒) ∧ 𝜓) ∧ 𝜑) → 𝜏)       ((((𝜑𝜓) ∧ 𝜒) ∧ 𝜃) → 𝜏)

Theoremexpcomdg 39023 Biconditional form of expcomd 453. (Contributed by Alan Sare, 22-Jul-2012.) (New usage is discouraged.)
((𝜑 → ((𝜓𝜒) → 𝜃)) ↔ (𝜑 → (𝜒 → (𝜓𝜃))))

20.31.3  Conventional Metamath proofs, some derived from VD proofs

Theoremiidn3 39024 idn3 39157 without virtual deduction connectives. Special theorem needed for the Virtual Deduction translation tool. (Contributed by Alan Sare, 23-Jul-2011.) (Proof modification is discouraged.) (New usage is discouraged.)
(𝜑 → (𝜓 → (𝜒𝜒)))

Theoremee222 39025 e222 39178 without virtual deduction connectives. Special theorem needed for the Virtual Deduction translation tool. (Contributed by Alan Sare, 7-Jul-2011.) (Proof modification is discouraged.) (New usage is discouraged.)
(𝜑 → (𝜓𝜒))    &   (𝜑 → (𝜓𝜃))    &   (𝜑 → (𝜓𝜏))    &   (𝜒 → (𝜃 → (𝜏𝜂)))       (𝜑 → (𝜓𝜂))

Theoremee3bir 39026 Right-biconditional form of e3 39281 without virtual deduction connectives. Special theorem needed for the Virtual Deduction translation tool. (Contributed by Alan Sare, 22-Jul-2011.) (Proof modification is discouraged.) (New usage is discouraged.)
(𝜑 → (𝜓 → (𝜒𝜃)))    &   (𝜏𝜃)       (𝜑 → (𝜓 → (𝜒𝜏)))

Theoremee13 39027 e13 39292 without virtual deduction connectives. Special theorem needed for the Virtual Deduction translation tool. (Contributed by Alan Sare, 28-Oct-2011.) (Proof modification is discouraged.) (New usage is discouraged.)
(𝜑𝜓)    &   (𝜑 → (𝜒 → (𝜃𝜏)))    &   (𝜓 → (𝜏𝜂))       (𝜑 → (𝜒 → (𝜃𝜂)))

Theoremee121 39028 e121 39198 without virtual deductions. (Contributed by Alan Sare, 13-Jul-2011.) (Proof modification is discouraged.) (New usage is discouraged.)
(𝜑𝜓)    &   (𝜑 → (𝜒𝜃))    &   (𝜑𝜏)    &   (𝜓 → (𝜃 → (𝜏𝜂)))       (𝜑 → (𝜒𝜂))

Theoremee122 39029 e122 39195 without virtual deductions. (Contributed by Alan Sare, 13-Jul-2011.) (Proof modification is discouraged.) (New usage is discouraged.)
(𝜑𝜓)    &   (𝜑 → (𝜒𝜃))    &   (𝜑 → (𝜒𝜏))    &   (𝜓 → (𝜃 → (𝜏𝜂)))       (𝜑 → (𝜒𝜂))

Theoremee333 39030 e333 39277 without virtual deductions. (Contributed by Alan Sare, 17-Jul-2011.) (Proof modification is discouraged.) (New usage is discouraged.)
(𝜑 → (𝜓 → (𝜒𝜃)))    &   (𝜑 → (𝜓 → (𝜒𝜏)))    &   (𝜑 → (𝜓 → (𝜒𝜂)))    &   (𝜃 → (𝜏 → (𝜂𝜁)))       (𝜑 → (𝜓 → (𝜒𝜁)))

Theoremee323 39031 e323 39310 without virtual deductions. (Contributed by Alan Sare, 17-Apr-2012.) (Proof modification is discouraged.) (New usage is discouraged.)
(𝜑 → (𝜓 → (𝜒𝜃)))    &   (𝜑 → (𝜓𝜏))    &   (𝜑 → (𝜓 → (𝜒𝜂)))    &   (𝜃 → (𝜏 → (𝜂𝜁)))       (𝜑 → (𝜓 → (𝜒𝜁)))

Theorem3ornot23 39032 If the second and third disjuncts of a true triple disjunction are false, then the first disjunct is true. Automatically derived from 3ornot23VD 39396. (Contributed by Alan Sare, 31-Dec-2011.) (Proof modification is discouraged.) (New usage is discouraged.)
((¬ 𝜑 ∧ ¬ 𝜓) → ((𝜒𝜑𝜓) → 𝜒))

Theoremorbi1r 39033 orbi1 742 with order of disjuncts reversed. Derived from orbi1rVD 39397. (Contributed by Alan Sare, 31-Dec-2011.) (Proof modification is discouraged.) (New usage is discouraged.)
((𝜑𝜓) → ((𝜒𝜑) ↔ (𝜒𝜓)))

Theorem3orbi123 39034 pm4.39 933 with a 3-conjunct antecedent. This proof is 3orbi123VD 39399 automatically translated and minimized. (Contributed by Alan Sare, 31-Dec-2011.) (Proof modification is discouraged.) (New usage is discouraged.)
(((𝜑𝜓) ∧ (𝜒𝜃) ∧ (𝜏𝜂)) → ((𝜑𝜒𝜏) ↔ (𝜓𝜃𝜂)))

Theoremsyl5imp 39035 Closed form of syl5 34. Derived automatically from syl5impVD 39413. (Contributed by Alan Sare, 31-Dec-2011.) (Proof modification is discouraged.) (New usage is discouraged.)
((𝜑 → (𝜓𝜒)) → ((𝜃𝜓) → (𝜑 → (𝜃𝜒))))

Theoremimpexpd 39036 The following User's Proof is a Virtual Deduction proof completed automatically by the tools program completeusersproof.cmd, which invokes Mel L. O'Cat's mmj2 and Norm Megill's Metamath Proof Assistant. After the User's Proof was completed, it was minimized. The completed User's Proof before minimization is not shown. (Contributed by Alan Sare, 18-Mar-2012.) (Proof modification is discouraged.) (New usage is discouraged.)
 1:: ⊢ (((𝜓 ∧ 𝜒) → 𝜃) ↔ (𝜓 → (𝜒 → 𝜃))) qed:1: ⊢ ((𝜑 → ((𝜓 ∧ 𝜒) → 𝜃)) ↔ (𝜑 → (𝜓 → (𝜒 → 𝜃))))
((𝜑 → ((𝜓𝜒) → 𝜃)) ↔ (𝜑 → (𝜓 → (𝜒𝜃))))

Theoremcom3rgbi 39037 The following User's Proof is a Virtual Deduction proof completed automatically by the tools program completeusersproof.cmd, which invokes Mel L. O'Cat's mmj2 and Norm Megill's Metamath Proof Assistant. The completed Virtual Deduction Proof (not shown) was minimized. The minimized proof is shown. (Contributed by Alan Sare, 18-Mar-2012.) (Proof modification is discouraged.) (New usage is discouraged.)
 1:: ⊢ ((𝜑 → (𝜓 → (𝜒 → 𝜃))) → (𝜑 → (𝜒 → (𝜓 → 𝜃)))) 2:: ⊢ ((𝜑 → (𝜒 → (𝜓 → 𝜃))) → (𝜒 → (𝜑 → (𝜓 → 𝜃)))) 3:1,2: ⊢ ((𝜑 → (𝜓 → (𝜒 → 𝜃))) → (𝜒 → (𝜑 → (𝜓 → 𝜃)))) 4:: ⊢ ((𝜒 → (𝜑 → (𝜓 → 𝜃))) → (𝜑 → (𝜒 → (𝜓 → 𝜃)))) 5:: ⊢ ((𝜑 → (𝜒 → (𝜓 → 𝜃))) → (𝜑 → (𝜓 → (𝜒 → 𝜃)))) 6:4,5: ⊢ ((𝜒 → (𝜑 → (𝜓 → 𝜃))) → (𝜑 → (𝜓 → (𝜒 → 𝜃)))) qed:3,6: ⊢ ((𝜑 → (𝜓 → (𝜒 → 𝜃))) ↔ (𝜒 → (𝜑 → (𝜓 → 𝜃))))
((𝜑 → (𝜓 → (𝜒𝜃))) ↔ (𝜒 → (𝜑 → (𝜓𝜃))))

Theoremimpexpdcom 39038 The following User's Proof is a Virtual Deduction proof completed automatically by the tools program completeusersproof.cmd, which invokes Mel L. O'Cat's mmj2 and Norm Megill's Metamath Proof Assistant. The completed Virtual Deduction Proof (not shown) was minimized. The minimized proof is shown. (Contributed by Alan Sare, 18-Mar-2012.) (Proof modification is discouraged.) (New usage is discouraged.)
 1:: ⊢ ((𝜑 → ((𝜓 ∧ 𝜒) → 𝜃)) ↔ (𝜑 → (𝜓 → (𝜒 → 𝜃)))) 2:: ⊢ ((𝜓 → (𝜒 → (𝜑 → 𝜃))) ↔ (𝜑 → (𝜓 → (𝜒 → 𝜃)))) qed:1,2: ⊢ ((𝜑 → ((𝜓 ∧ 𝜒) → 𝜃)) ↔ (𝜓 → (𝜒 → (𝜑 → 𝜃))))
((𝜑 → ((𝜓𝜒) → 𝜃)) ↔ (𝜓 → (𝜒 → (𝜑𝜃))))

Theoremee1111 39039 Non-virtual deduction form of e1111 39217. (Contributed by Alan Sare, 18-Mar-2012.) (Proof modification is discouraged.) (New usage is discouraged.) The following User's Proof is a Virtual Deduction proof completed automatically by the tools program completeusersproof.cmd, which invokes Mel L. O'Cat's mmj2 and Norm Megill's Metamath Proof Assistant. The completed Virtual Deduction Proof (not shown) was minimized. The minimized proof is shown.
 h1:: ⊢ (𝜑 → 𝜓) h2:: ⊢ (𝜑 → 𝜒) h3:: ⊢ (𝜑 → 𝜃) h4:: ⊢ (𝜑 → 𝜏) h5:: ⊢ (𝜓 → (𝜒 → (𝜃 → (𝜏 → 𝜂)))) 6:1,5: ⊢ (𝜑 → (𝜒 → (𝜃 → (𝜏 → 𝜂)))) 7:6: ⊢ (𝜒 → (𝜑 → (𝜃 → (𝜏 → 𝜂)))) 8:2,7: ⊢ (𝜑 → (𝜑 → (𝜃 → (𝜏 → 𝜂)))) 9:8: ⊢ (𝜑 → (𝜃 → (𝜏 → 𝜂))) 10:9: ⊢ (𝜃 → (𝜑 → (𝜏 → 𝜂))) 11:3,10: ⊢ (𝜑 → (𝜑 → (𝜏 → 𝜂))) 12:11: ⊢ (𝜑 → (𝜏 → 𝜂)) 13:12: ⊢ (𝜏 → (𝜑 → 𝜂)) 14:4,13: ⊢ (𝜑 → (𝜑 → 𝜂)) qed:14: ⊢ (𝜑 → 𝜂)
(𝜑𝜓)    &   (𝜑𝜒)    &   (𝜑𝜃)    &   (𝜑𝜏)    &   (𝜓 → (𝜒 → (𝜃 → (𝜏𝜂))))       (𝜑𝜂)

Theorempm2.43bgbi 39040 Logical equivalence of a 2-left-nested implication and a 1-left-nested implicated when two antecedents of the former implication are identical. (Contributed by Alan Sare, 18-Mar-2012.) (Proof modification is discouraged.) (New usage is discouraged.) The following User's Proof is a Virtual Deduction proof completed automatically by the tools program completeusersproof.cmd, which invokes Mel L. O'Cat's mmj2 and Norm Megill's Metamath Proof Assistant. The completed Virtual Deduction Proof (not shown) was minimized. The minimized proof is shown.
 1:: ⊢ ((𝜑 → (𝜓 → (𝜑 → 𝜒))) → (𝜑 → (𝜑 → (𝜓 → 𝜒)))) 2:: ⊢ ((𝜑 → (𝜑 → (𝜓 → 𝜒))) → (𝜑 → (𝜓 → 𝜒))) 3:1,2: ⊢ ((𝜑 → (𝜓 → (𝜑 → 𝜒))) → (𝜑 → (𝜓 → 𝜒))) 4:: ⊢ ((𝜑 → (𝜓 → 𝜒)) → (𝜓 → (𝜑 → 𝜒))) 5:3,4: ⊢ ((𝜑 → (𝜓 → (𝜑 → 𝜒))) → (𝜓 → (𝜑 → 𝜒))) 6:: ⊢ ((𝜓 → (𝜑 → 𝜒)) → (𝜑 → (𝜓 → (𝜑 → 𝜒)))) qed:5,6: ⊢ ((𝜑 → (𝜓 → (𝜑 → 𝜒))) ↔ (𝜓 → (𝜑 → 𝜒)))
((𝜑 → (𝜓 → (𝜑𝜒))) ↔ (𝜓 → (𝜑𝜒)))

Theorempm2.43cbi 39041 Logical equivalence of a 3-left-nested implication and a 2-left-nested implicated when two antecedents of the former implication are identical. (Contributed by Alan Sare, 18-Mar-2012.) (Proof modification is discouraged.) (New usage is discouraged.) The following User's Proof is a Virtual Deduction proof completed automatically by the tools program completeusersproof.cmd, which invokes Mel L. O'Cat's mmj2 and Norm Megill's Metamath Proof Assistant. The completed Virtual Deduction Proof (not shown) was minimized. The minimized proof is shown.
 1:: ⊢ ((𝜑 → (𝜓 → (𝜒 → (𝜑 → 𝜃))) ) → (𝜑 → (𝜓 → (𝜑 → (𝜒 → 𝜃))))) 2:: ⊢ ((𝜑 → (𝜓 → (𝜑 → (𝜒 → 𝜃))) ) → (𝜓 → (𝜑 → (𝜒 → 𝜃)))) 3:1,2: ⊢ ((𝜑 → (𝜓 → (𝜒 → (𝜑 → 𝜃))) ) → (𝜓 → (𝜑 → (𝜒 → 𝜃)))) 4:: ⊢ ((𝜓 → (𝜑 → (𝜒 → 𝜃))) → (𝜓 → (𝜒 → (𝜑 → 𝜃)))) 5:3,4: ⊢ ((𝜑 → (𝜓 → (𝜒 → (𝜑 → 𝜃))) ) → (𝜓 → (𝜒 → (𝜑 → 𝜃)))) 6:: ⊢ ((𝜓 → (𝜒 → (𝜑 → 𝜃))) → (𝜑 → (𝜓 → (𝜒 → (𝜑 → 𝜃))))) qed:5,6: ⊢ ((𝜑 → (𝜓 → (𝜒 → (𝜑 → 𝜃))) ) ↔ (𝜓 → (𝜒 → (𝜑 → 𝜃))))
((𝜑 → (𝜓 → (𝜒 → (𝜑𝜃)))) ↔ (𝜓 → (𝜒 → (𝜑𝜃))))

Theoremee233 39042 Non-virtual deduction form of e233 39309. (Contributed by Alan Sare, 18-Mar-2012.) (Proof modification is discouraged.) (New usage is discouraged.) The following User's Proof is a Virtual Deduction proof completed automatically by the tools program completeusersproof.cmd, which invokes Mel L. O'Cat's mmj2 and Norm Megill's Metamath Proof Assistant. The completed Virtual Deduction Proof (not shown) was minimized. The minimized proof is shown.
 h1:: ⊢ (𝜑 → (𝜓 → 𝜒)) h2:: ⊢ (𝜑 → (𝜓 → (𝜃 → 𝜏))) h3:: ⊢ (𝜑 → (𝜓 → (𝜃 → 𝜂))) h4:: ⊢ (𝜒 → (𝜏 → (𝜂 → 𝜁))) 5:1,4: ⊢ (𝜑 → (𝜓 → (𝜏 → (𝜂 → 𝜁))) ) 6:5: ⊢ (𝜏 → (𝜑 → (𝜓 → (𝜂 → 𝜁))) ) 7:2,6: ⊢ (𝜑 → (𝜓 → (𝜃 → (𝜑 → (𝜓 → (𝜂 → 𝜁)))))) 8:7: ⊢ (𝜓 → (𝜃 → (𝜑 → (𝜓 → (𝜂 → 𝜁))))) 9:8: ⊢ (𝜃 → (𝜑 → (𝜓 → (𝜂 → 𝜁))) ) 10:9: ⊢ (𝜑 → (𝜓 → (𝜃 → (𝜂 → 𝜁))) ) 11:10: ⊢ (𝜂 → (𝜑 → (𝜓 → (𝜃 → 𝜁))) ) 12:3,11: ⊢ (𝜑 → (𝜓 → (𝜃 → (𝜑 → (𝜓 → (𝜃 → 𝜁)))))) 13:12: ⊢ (𝜓 → (𝜃 → (𝜑 → (𝜓 → (𝜃 → 𝜁))))) 14:13: ⊢ (𝜃 → (𝜑 → (𝜓 → (𝜃 → 𝜁))) ) qed:14: ⊢ (𝜑 → (𝜓 → (𝜃 → 𝜁)))
(𝜑 → (𝜓𝜒))    &   (𝜑 → (𝜓 → (𝜃𝜏)))    &   (𝜑 → (𝜓 → (𝜃𝜂)))    &   (𝜒 → (𝜏 → (𝜂𝜁)))       (𝜑 → (𝜓 → (𝜃𝜁)))

Theoremimbi13 39043 Join three logical equivalences to form equivalence of implications. imbi13 39043 is imbi13VD 39424 without virtual deductions and was automatically derived from imbi13VD 39424 using the tools program translate..without..overwriting.cmd and Metamath's minimize command. (Contributed by Alan Sare, 18-Mar-2012.) (Proof modification is discouraged.) (New usage is discouraged.)
((𝜑𝜓) → ((𝜒𝜃) → ((𝜏𝜂) → ((𝜑 → (𝜒𝜏)) ↔ (𝜓 → (𝜃𝜂))))))

Theoremee33 39044 Non-virtual deduction form of e33 39278. (Contributed by Alan Sare, 18-Mar-2012.) (Proof modification is discouraged.) (New usage is discouraged.) The following User's Proof is a Virtual Deduction proof completed automatically by the tools program completeusersproof.cmd, which invokes Mel L. O'Cat's mmj2 and Norm Megill's Metamath Proof Assistant. The completed Virtual Deduction Proof (not shown) was minimized. The minimized proof is shown.
 h1:: ⊢ (𝜑 → (𝜓 → (𝜒 → 𝜃))) h2:: ⊢ (𝜑 → (𝜓 → (𝜒 → 𝜏))) h3:: ⊢ (𝜃 → (𝜏 → 𝜂)) 4:1,3: ⊢ (𝜑 → (𝜓 → (𝜒 → (𝜏 → 𝜂)))) 5:4: ⊢ (𝜏 → (𝜑 → (𝜓 → (𝜒 → 𝜂)))) 6:2,5: ⊢ (𝜑 → (𝜓 → (𝜒 → (𝜑 → (𝜓 → (𝜒 → 𝜂)))))) 7:6: ⊢ (𝜓 → (𝜒 → (𝜑 → (𝜓 → (𝜒 → 𝜂))))) 8:7: ⊢ (𝜒 → (𝜑 → (𝜓 → (𝜒 → 𝜂)))) qed:8: ⊢ (𝜑 → (𝜓 → (𝜒 → 𝜂)))
(𝜑 → (𝜓 → (𝜒𝜃)))    &   (𝜑 → (𝜓 → (𝜒𝜏)))    &   (𝜃 → (𝜏𝜂))       (𝜑 → (𝜓 → (𝜒𝜂)))

Theoremcon5 39045 Biconditional contraposition variation. This proof is con5VD 39450 automatically translated and minimized. (Contributed by Alan Sare, 21-Apr-2013.) (Proof modification is discouraged.) (New usage is discouraged.)
((𝜑 ↔ ¬ 𝜓) → (¬ 𝜑𝜓))

Theoremcon5i 39046 Inference form of con5 39045. (Contributed by Alan Sare, 21-Apr-2013.) (Proof modification is discouraged.) (New usage is discouraged.)
(𝜑 ↔ ¬ 𝜓)       𝜑𝜓)

Theoremexlimexi 39047 Inference similar to Theorem 19.23 of [Margaris] p. 90. (Contributed by Alan Sare, 21-Apr-2013.) (Proof modification is discouraged.) (New usage is discouraged.)
(𝜓 → ∀𝑥𝜓)    &   (∃𝑥𝜑 → (𝜑𝜓))       (∃𝑥𝜑𝜓)

Theoremsb5ALT 39048* Equivalence for substitution. Alternate proof of sb5 2458. This proof is sb5ALTVD 39463 automatically translated and minimized. (Contributed by Alan Sare, 21-Apr-2013.) (Proof modification is discouraged.) (New usage is discouraged.)
([𝑦 / 𝑥]𝜑 ↔ ∃𝑥(𝑥 = 𝑦𝜑))

Theoremeexinst01 39049 exinst01 39167 without virtual deductions. (Contributed by Alan Sare, 21-Apr-2013.) (Proof modification is discouraged.) (New usage is discouraged.)
𝑥𝜓    &   (𝜑 → (𝜓𝜒))    &   (𝜑 → ∀𝑥𝜑)    &   (𝜒 → ∀𝑥𝜒)       (𝜑𝜒)

Theoremeexinst11 39050 exinst11 39168 without virtual deductions. (Contributed by Alan Sare, 21-Apr-2013.) (Proof modification is discouraged.) (New usage is discouraged.)
(𝜑 → ∃𝑥𝜓)    &   (𝜑 → (𝜓𝜒))    &   (𝜑 → ∀𝑥𝜑)    &   (𝜒 → ∀𝑥𝜒)       (𝜑𝜒)

Theoremvk15.4j 39051 Excercise 4j of Unit 15 of "Understanding Symbolic Logic", Fifth Edition (2008), by Virginia Klenk. This proof is the minimized Hilbert-style axiomatic version of the Fitch-style Natural Deduction proof found on page 442 of Klenk and was automatically derived from that proof. vk15.4j 39051 is vk15.4jVD 39464 automatically translated and minimized. (Contributed by Alan Sare, 21-Apr-2013.) (Proof modification is discouraged.) (New usage is discouraged.)
¬ (∃𝑥 ¬ 𝜑 ∧ ∃𝑥(𝜓 ∧ ¬ 𝜒))    &   (∀𝑥𝜒 → ¬ ∃𝑥(𝜃𝜏))    &    ¬ ∀𝑥(𝜏𝜑)       (¬ ∃𝑥 ¬ 𝜃 → ¬ ∀𝑥𝜓)

TheoremnotnotrALT 39052 Converse of double negation. Alternate proof of notnotr 125. This proof is notnotrALTVD 39465 automatically translated and minimized. (Contributed by Alan Sare, 21-Apr-2013.) (Proof modification is discouraged.) (New usage is discouraged.)
(¬ ¬ 𝜑𝜑)

Theoremcon3ALT2 39053 Contraposition. Alternate proof of con3 149. This proof is con3ALTVD 39466 automatically translated and minimized. (Contributed by Alan Sare, 21-Apr-2013.) (Proof modification is discouraged.) (New usage is discouraged.)
((𝜑𝜓) → (¬ 𝜓 → ¬ 𝜑))

Theoremssralv2 39054* Quantification restricted to a subclass for two quantifiers. ssralv 3699 for two quantifiers. The proof of ssralv2 39054 was automatically generated by minimizing the automatically translated proof of ssralv2VD 39416. The automatic translation is by the tools program translatewithout_overwriting.cmd. (Contributed by Alan Sare, 18-Feb-2012.) (Proof modification is discouraged.) (New usage is discouraged.)
((𝐴𝐵𝐶𝐷) → (∀𝑥𝐵𝑦𝐷 𝜑 → ∀𝑥𝐴𝑦𝐶 𝜑))

Theoremsbc3or 39055 sbcor 3512 with a 3-disjuncts. This proof is sbc3orgVD 39400 automatically translated and minimized. (Contributed by Alan Sare, 31-Dec-2011.) (Revised by NM, 24-Aug-2018.) (Proof modification is discouraged.) (New usage is discouraged.)
([𝐴 / 𝑥](𝜑𝜓𝜒) ↔ ([𝐴 / 𝑥]𝜑[𝐴 / 𝑥]𝜓[𝐴 / 𝑥]𝜒))

TheoremsbcangOLD 39056 Distribution of class substitution over conjunction. (Contributed by NM, 21-May-2004.) Obsolete as of 17-Aug-2018. Use sbcan 3511 instead. (New usage is discouraged.) (Proof modification is discouraged.)
(𝐴𝑉 → ([𝐴 / 𝑥](𝜑𝜓) ↔ ([𝐴 / 𝑥]𝜑[𝐴 / 𝑥]𝜓)))

TheoremsbcorgOLD 39057 Distribution of class substitution over disjunction. (Contributed by NM, 21-May-2004.) Obsolete as of 17-Aug-2018. Use sbcor 3512 instead. (New usage is discouraged.) (Proof modification is discouraged.)
(𝐴𝑉 → ([𝐴 / 𝑥](𝜑𝜓) ↔ ([𝐴 / 𝑥]𝜑[𝐴 / 𝑥]𝜓)))

TheoremsbcbiiOLD 39058 Formula-building inference rule for class substitution. (Contributed by NM, 11-Nov-2005.) Obsolete as of 17-Aug-2018. Use sbcbii 3524 instead. (New usage is discouraged.) (Proof modification is discouraged.)
(𝜑𝜓)       (𝐴𝑉 → ([𝐴 / 𝑥]𝜑[𝐴 / 𝑥]𝜓))

Theoremsbc3orgOLD 39059 sbcorgOLD 39057 with a 3-disjuncts. This proof is sbc3orgVD 39400 automatically translated and minimized. (Contributed by Alan Sare, 31-Dec-2011.) (Proof modification is discouraged.) (New usage is discouraged.)
(𝐴𝑉 → ([𝐴 / 𝑥](𝜑𝜓𝜒) ↔ ([𝐴 / 𝑥]𝜑[𝐴 / 𝑥]𝜓[𝐴 / 𝑥]𝜒)))

Theoremalrim3con13v 39060* Closed form of alrimi 2120 with 2 additional conjuncts having no occurrences of the quantifying variable. This proof is 19.21a3con13vVD 39401 automatically translated and minimized. (Contributed by Alan Sare, 31-Dec-2011.) (Proof modification is discouraged.) (New usage is discouraged.)
((𝜑 → ∀𝑥𝜑) → ((𝜓𝜑𝜒) → ∀𝑥(𝜓𝜑𝜒)))

Theoremrspsbc2 39061* rspsbc 3551 with two quantifying variables. This proof is rspsbc2VD 39404 automatically translated and minimized. (Contributed by Alan Sare, 31-Dec-2011.) (Proof modification is discouraged.) (New usage is discouraged.)
(𝐴𝐵 → (𝐶𝐷 → (∀𝑥𝐵𝑦𝐷 𝜑[𝐶 / 𝑦][𝐴 / 𝑥]𝜑)))

Theoremsbcoreleleq 39062* Substitution of a setvar variable for another setvar variable in a 3-conjunct formula. Derived automatically from sbcoreleleqVD 39409. (Contributed by Alan Sare, 31-Dec-2011.) (Proof modification is discouraged.) (New usage is discouraged.)
(𝐴𝑉 → ([𝐴 / 𝑦](𝑥𝑦𝑦𝑥𝑥 = 𝑦) ↔ (𝑥𝐴𝐴𝑥𝑥 = 𝐴)))

Theoremtratrb 39063* If a class is transitive and any two distinct elements of the class are E-comparable, then every element of that class is transitive. Derived automatically from tratrbVD 39411. (Contributed by Alan Sare, 31-Dec-2011.) (Proof modification is discouraged.) (New usage is discouraged.)
((Tr 𝐴 ∧ ∀𝑥𝐴𝑦𝐴 (𝑥𝑦𝑦𝑥𝑥 = 𝑦) ∧ 𝐵𝐴) → Tr 𝐵)

TheoremordelordALT 39064 An element of an ordinal class is ordinal. Proposition 7.6 of [TakeutiZaring] p. 36. This is an alternate proof of ordelord 5783 using the Axiom of Regularity indirectly through dford2 8555. dford2 is a weaker definition of ordinal number. Given the Axiom of Regularity, it need not be assumed that E Fr 𝐴 because this is inferred by the Axiom of Regularity. ordelordALT 39064 is ordelordALTVD 39417 without virtual deductions and was automatically derived from ordelordALTVD 39417 using the tools program translate..without..overwriting.cmd and Metamath's minimize command. (Contributed by Alan Sare, 18-Feb-2012.) (Proof modification is discouraged.) (New usage is discouraged.)
((Ord 𝐴𝐵𝐴) → Ord 𝐵)

Theoremsbcim2g 39065 Distribution of class substitution over a left-nested implication. Similar to sbcimg 3510. sbcim2g 39065 is sbcim2gVD 39425 without virtual deductions and was automatically derived from sbcim2gVD 39425 using the tools program translate..without..overwriting.cmd and Metamath's minimize command. (Contributed by Alan Sare, 18-Mar-2012.) (Proof modification is discouraged.) (New usage is discouraged.)
(𝐴𝑉 → ([𝐴 / 𝑥](𝜑 → (𝜓𝜒)) ↔ ([𝐴 / 𝑥]𝜑 → ([𝐴 / 𝑥]𝜓[𝐴 / 𝑥]𝜒))))

Theoremsbcbi 39066 Implication form of sbcbiiOLD 39058. sbcbi 39066 is sbcbiVD 39426 without virtual deductions and was automatically derived from sbcbiVD 39426 using the tools program translate..without..overwriting.cmd and Metamath's minimize command. (Contributed by Alan Sare, 18-Mar-2012.) (Proof modification is discouraged.) (New usage is discouraged.)
(𝐴𝑉 → (∀𝑥(𝜑𝜓) → ([𝐴 / 𝑥]𝜑[𝐴 / 𝑥]𝜓)))

Theoremtrsbc 39067* Formula-building inference rule for class substitution, substituting a class variable for the setvar variable of the transitivity predicate. trsbc 39067 is trsbcVD 39427 without virtual deductions and was automatically derived from trsbcVD 39427 using the tools program translate..without..overwriting.cmd and Metamath's minimize command. (Contributed by Alan Sare, 18-Mar-2012.) (Proof modification is discouraged.) (New usage is discouraged.)
(𝐴𝑉 → ([𝐴 / 𝑥]Tr 𝑥 ↔ Tr 𝐴))

TheoremtruniALT 39068* The union of a class of transitive sets is transitive. Alternate proof of truni 4800. truniALT 39068 is truniALTVD 39428 without virtual deductions and was automatically derived from truniALTVD 39428 using the tools program translate..without..overwriting.cmd and Metamath's minimize command. (Contributed by Alan Sare, 18-Mar-2012.) (Proof modification is discouraged.) (New usage is discouraged.)
(∀𝑥𝐴 Tr 𝑥 → Tr 𝐴)

TheoremsbcalgOLD 39069* Move universal quantifier in and out of class substitution. (Contributed by NM, 16-Jan-2004.) Obsolete as of 17-Aug-2018. Use sbcal 3518 instead. (New usage is discouraged.) (Proof modification is discouraged.)
(𝐴𝑉 → ([𝐴 / 𝑦]𝑥𝜑 ↔ ∀𝑥[𝐴 / 𝑦]𝜑))

TheoremsbcexgOLD 39070* Move existential quantifier in and out of class substitution. (Contributed by NM, 21-May-2004.) Obsolete as of 17-Aug-2018. Use sbcex2 3519 instead. (New usage is discouraged.) (Proof modification is discouraged.)
(𝐴𝑉 → ([𝐴 / 𝑦]𝑥𝜑 ↔ ∃𝑥[𝐴 / 𝑦]𝜑))

Theoremsbcel12gOLD 39071 Distribute proper substitution through a membership relation. (Contributed by NM, 10-Nov-2005.) (Proof shortened by Andrew Salmon, 29-Jun-2011.) Obsolete as of 18-Aug-2018. Use sbcel12 4016 instead. (New usage is discouraged.) (Proof modification is discouraged.)
(𝐴𝑉 → ([𝐴 / 𝑥]𝐵𝐶𝐴 / 𝑥𝐵𝐴 / 𝑥𝐶))

Theoremsbcel2gOLD 39072* Move proper substitution in and out of a membership relation. (Contributed by NM, 14-Nov-2005.) Obsolete as of 18-Aug-2018. Use sbcel2 4022 instead. (New usage is discouraged.) (Proof modification is discouraged.)
(𝐴𝑉 → ([𝐴 / 𝑥]𝐵𝐶𝐵𝐴 / 𝑥𝐶))

TheoremsbcssOLD 39073 Distribute proper substitution through a subclass relation. This theorem was automatically derived from sbcssgVD 39433. (Contributed by Alan Sare, 22-Jul-2012.) (Proof modification is discouraged.) (New usage is discouraged.)
(𝐴𝐵 → ([𝐴 / 𝑥]𝐶𝐷𝐴 / 𝑥𝐶𝐴 / 𝑥𝐷))

TheoremonfrALTlem5 39074* Lemma for onfrALT 39081. (Contributed by Alan Sare, 22-Jul-2012.) (Proof modification is discouraged.) (New usage is discouraged.)
([(𝑎𝑥) / 𝑏]((𝑏 ⊆ (𝑎𝑥) ∧ 𝑏 ≠ ∅) → ∃𝑦𝑏 (𝑏𝑦) = ∅) ↔ (((𝑎𝑥) ⊆ (𝑎𝑥) ∧ (𝑎𝑥) ≠ ∅) → ∃𝑦 ∈ (𝑎𝑥)((𝑎𝑥) ∩ 𝑦) = ∅))

TheoremonfrALTlem4 39075* Lemma for onfrALT 39081. (Contributed by Alan Sare, 22-Jul-2012.) (Proof modification is discouraged.) (New usage is discouraged.)
([𝑦 / 𝑥](𝑥𝑎 ∧ (𝑎𝑥) = ∅) ↔ (𝑦𝑎 ∧ (𝑎𝑦) = ∅))

TheoremonfrALTlem3 39076* Lemma for onfrALT 39081. (Contributed by Alan Sare, 22-Jul-2012.) (Proof modification is discouraged.) (New usage is discouraged.)
((𝑎 ⊆ On ∧ 𝑎 ≠ ∅) → ((𝑥𝑎 ∧ ¬ (𝑎𝑥) = ∅) → ∃𝑦 ∈ (𝑎𝑥)((𝑎𝑥) ∩ 𝑦) = ∅))

Theoremggen31 39077* gen31 39163 without virtual deductions. (Contributed by Alan Sare, 22-Jul-2012.) (Proof modification is discouraged.) (New usage is discouraged.)
(𝜑 → (𝜓 → (𝜒𝜃)))       (𝜑 → (𝜓 → (𝜒 → ∀𝑥𝜃)))

TheoremonfrALTlem2 39078* Lemma for onfrALT 39081. (Contributed by Alan Sare, 22-Jul-2012.) (Proof modification is discouraged.) (New usage is discouraged.)
((𝑎 ⊆ On ∧ 𝑎 ≠ ∅) → ((𝑥𝑎 ∧ ¬ (𝑎𝑥) = ∅) → ∃𝑦𝑎 (𝑎𝑦) = ∅))

Theoremcbvexsv 39079* A theorem pertaining to the substitution for an existentially quantified variable when the substituted variable does not occur in the quantified wff. (Contributed by Alan Sare, 22-Jul-2012.) (Proof modification is discouraged.) (New usage is discouraged.)
(∃𝑥𝜑 ↔ ∃𝑦[𝑦 / 𝑥]𝜑)

TheoremonfrALTlem1 39080* Lemma for onfrALT 39081. (Contributed by Alan Sare, 22-Jul-2012.) (Proof modification is discouraged.) (New usage is discouraged.)
((𝑎 ⊆ On ∧ 𝑎 ≠ ∅) → ((𝑥𝑎 ∧ (𝑎𝑥) = ∅) → ∃𝑦𝑎 (𝑎𝑦) = ∅))

TheoremonfrALT 39081 The epsilon relation is foundational on the class of ordinal numbers. onfrALT 39081 is an alternate proof of onfr 5801. onfrALTVD 39441 is the Virtual Deduction proof from which onfrALT 39081 is derived. The Virtual Deduction proof mirrors the working proof of onfr 5801 which is the main part of the proof of Theorem 7.12 of the first edition of TakeutiZaring. The proof of the corresponding Proposition 7.12 of [TakeutiZaring] p. 38 (second edition) does not contain the working proof equivalent of onfrALTVD 39441. This theorem does not rely on the Axiom of Regularity. (Contributed by Alan Sare, 22-Jul-2012.) (Proof modification is discouraged.) (New usage is discouraged.)
E Fr On

Theoremcsbeq2gOLD 39082 Formula-building implication rule for class substitution. Closed form of csbeq2i 4026. csbeq2gOLD 39082 is derived from the virtual deduction proof csbeq2gVD 39442. (Contributed by Alan Sare, 10-Nov-2012.) Obsolete version of csbeq2 3570 as of 11-Oct-2018. (Proof modification is discouraged.) (New usage is discouraged.)
(𝐴𝑉 → (∀𝑥 𝐵 = 𝐶𝐴 / 𝑥𝐵 = 𝐴 / 𝑥𝐶))

Theorem19.41rg 39083 Closed form of right-to-left implication of 19.41 2141, Theorem 19.41 of [Margaris] p. 90. Derived from 19.41rgVD 39452. (Contributed by Alan Sare, 8-Feb-2014.) (Proof modification is discouraged.) (New usage is discouraged.)
(∀𝑥(𝜓 → ∀𝑥𝜓) → ((∃𝑥𝜑𝜓) → ∃𝑥(𝜑𝜓)))

Theoremopelopab4 39084* Ordered pair membership in a class abstraction of pairs. Compare to elopab 5012. (Contributed by Alan Sare, 8-Feb-2014.) (Revised by Mario Carneiro, 6-May-2015.) (Proof modification is discouraged.) (New usage is discouraged.)
(⟨𝑢, 𝑣⟩ ∈ {⟨𝑥, 𝑦⟩ ∣ 𝜑} ↔ ∃𝑥𝑦((𝑥 = 𝑢𝑦 = 𝑣) ∧ 𝜑))

Theorem2pm13.193 39085 pm13.193 38929 for two variables. pm13.193 38929 is Theorem *13.193 in [WhiteheadRussell] p. 179. Derived from 2pm13.193VD 39453. (Contributed by Alan Sare, 8-Feb-2014.) (Proof modification is discouraged.) (New usage is discouraged.)
(((𝑥 = 𝑢𝑦 = 𝑣) ∧ [𝑢 / 𝑥][𝑣 / 𝑦]𝜑) ↔ ((𝑥 = 𝑢𝑦 = 𝑣) ∧ 𝜑))

Theoremhbntal 39086 A closed form of hbn 2184. hbnt 2182 is another closed form of hbn 2184. (Contributed by Alan Sare, 8-Feb-2014.) (Proof modification is discouraged.) (New usage is discouraged.)
(∀𝑥(𝜑 → ∀𝑥𝜑) → ∀𝑥𝜑 → ∀𝑥 ¬ 𝜑))

Theoremhbimpg 39087 A closed form of hbim 2165. Derived from hbimpgVD 39454. (Contributed by Alan Sare, 8-Feb-2014.) (Proof modification is discouraged.) (New usage is discouraged.)
((∀𝑥(𝜑 → ∀𝑥𝜑) ∧ ∀𝑥(𝜓 → ∀𝑥𝜓)) → ∀𝑥((𝜑𝜓) → ∀𝑥(𝜑𝜓)))

Theoremhbalg 39088 Closed form of hbal 2076. Derived from hbalgVD 39455. (Contributed by Alan Sare, 8-Feb-2014.) (Proof modification is discouraged.) (New usage is discouraged.)
(∀𝑦(𝜑 → ∀𝑥𝜑) → ∀𝑦(∀𝑦𝜑 → ∀𝑥𝑦𝜑))

Theoremhbexg 39089 Closed form of nfex 2192. Derived from hbexgVD 39456. (Contributed by Alan Sare, 8-Feb-2014.) (Revised by Mario Carneiro, 12-Dec-2016.) (Proof modification is discouraged.) (New usage is discouraged.)
(∀𝑥𝑦(𝜑 → ∀𝑥𝜑) → ∀𝑥𝑦(∃𝑦𝜑 → ∀𝑥𝑦𝜑))

Theoremax6e2eq 39090* Alternate form of ax6e 2286 for non-distinct 𝑥, 𝑦 and 𝑢 = 𝑣. ax6e2eq 39090 is derived from ax6e2eqVD 39457. (Contributed by Alan Sare, 25-Mar-2014.) (Proof modification is discouraged.) (New usage is discouraged.)
(∀𝑥 𝑥 = 𝑦 → (𝑢 = 𝑣 → ∃𝑥𝑦(𝑥 = 𝑢𝑦 = 𝑣)))

Theoremax6e2nd 39091* If at least two sets exist (dtru 4887) , then the same is true expressed in an alternate form similar to the form of ax6e 2286. ax6e2nd 39091 is derived from ax6e2ndVD 39458. (Contributed by Alan Sare, 25-Mar-2014.) (Proof modification is discouraged.) (New usage is discouraged.)
(¬ ∀𝑥 𝑥 = 𝑦 → ∃𝑥𝑦(𝑥 = 𝑢𝑦 = 𝑣))

Theoremax6e2ndeq 39092* "At least two sets exist" expressed in the form of dtru 4887 is logically equivalent to the same expressed in a form similar to ax6e 2286 if dtru 4887 is false implies 𝑢 = 𝑣. ax6e2ndeq 39092 is derived from ax6e2ndeqVD 39459. (Contributed by Alan Sare, 25-Mar-2014.) (Proof modification is discouraged.) (New usage is discouraged.)
((¬ ∀𝑥 𝑥 = 𝑦𝑢 = 𝑣) ↔ ∃𝑥𝑦(𝑥 = 𝑢𝑦 = 𝑣))

Theorem2sb5nd 39093* Equivalence for double substitution 2sb5 2471 without distinct 𝑥, 𝑦 requirement. 2sb5nd 39093 is derived from 2sb5ndVD 39460. (Contributed by Alan Sare, 30-Apr-2014.) (Proof modification is discouraged.) (New usage is discouraged.)
((¬ ∀𝑥 𝑥 = 𝑦𝑢 = 𝑣) → ([𝑢 / 𝑥][𝑣 / 𝑦]𝜑 ↔ ∃𝑥𝑦((𝑥 = 𝑢𝑦 = 𝑣) ∧ 𝜑)))

Theorem2uasbanh 39094* Distribute the unabbreviated form of proper substitution in and out of a conjunction. 2uasbanh 39094 is derived from 2uasbanhVD 39461. (Contributed by Alan Sare, 31-May-2014.) (Proof modification is discouraged.) (New usage is discouraged.)
(𝜒 ↔ (∃𝑥𝑦((𝑥 = 𝑢𝑦 = 𝑣) ∧ 𝜑) ∧ ∃𝑥𝑦((𝑥 = 𝑢𝑦 = 𝑣) ∧ 𝜓)))       (∃𝑥𝑦((𝑥 = 𝑢𝑦 = 𝑣) ∧ (𝜑𝜓)) ↔ (∃𝑥𝑦((𝑥 = 𝑢𝑦 = 𝑣) ∧ 𝜑) ∧ ∃𝑥𝑦((𝑥 = 𝑢𝑦 = 𝑣) ∧ 𝜓)))

Theorem2uasban 39095* Distribute the unabbreviated form of proper substitution in and out of a conjunction. (Contributed by Alan Sare, 31-May-2014.) (Proof modification is discouraged.) (New usage is discouraged.)
(∃𝑥𝑦((𝑥 = 𝑢𝑦 = 𝑣) ∧ (𝜑𝜓)) ↔ (∃𝑥𝑦((𝑥 = 𝑢𝑦 = 𝑣) ∧ 𝜑) ∧ ∃𝑥𝑦((𝑥 = 𝑢𝑦 = 𝑣) ∧ 𝜓)))

Theoreme2ebind 39096 Absorption of an existential quantifier of a double existential quantifier of non-distinct variables. e2ebind 39096 is derived from e2ebindVD 39462. (Contributed by Alan Sare, 27-Nov-2014.) (Proof modification is discouraged.) (New usage is discouraged.)
(∀𝑥 𝑥 = 𝑦 → (∃𝑥𝑦𝜑 ↔ ∃𝑦𝜑))

Theoremelpwgded 39097 elpwgdedVD 39467 in conventional notation. (Contributed by Alan Sare, 23-Apr-2015.) (Proof modification is discouraged.) (New usage is discouraged.)
(𝜑𝐴 ∈ V)    &   (𝜓𝐴𝐵)       ((𝜑𝜓) → 𝐴 ∈ 𝒫 𝐵)

Theoremtrelded 39098 Deduction form of trel 4792. In a transitive class, the membership relation is transitive. (Contributed by Alan Sare, 3-Dec-2015.) (Proof modification is discouraged.) (New usage is discouraged.)
(𝜑 → Tr 𝐴)    &   (𝜓𝐵𝐶)    &   (𝜒𝐶𝐴)       ((𝜑𝜓𝜒) → 𝐵𝐴)

Theoremjaoded 39099 Deduction form of jao 533. Disjunction of antecedents. (Contributed by Alan Sare, 3-Dec-2015.) (Proof modification is discouraged.) (New usage is discouraged.)
(𝜑 → (𝜓𝜒))    &   (𝜃 → (𝜏𝜒))    &   (𝜂 → (𝜓𝜏))       ((𝜑𝜃𝜂) → 𝜒)

TheoremsbtT 39100 A substitution into a theorem remains true. sbt 2447 with the existence of no virtual hypotheses for the hypothesis expressed as the empty virtual hypothesis collection. (Contributed by Alan Sare, 4-Feb-2017.) (Proof modification is discouraged.) (New usage is discouraged.)
(⊤ → 𝜑)       [𝑦 / 𝑥]𝜑

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