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Theorem List for Metamath Proof Explorer - 2401-2500   *Has distinct variable group(s)
TypeLabelDescription
Statement

Theoremspime 2401 Existential introduction, using implicit substitution. Compare Lemma 14 of [Tarski] p. 70. (Contributed by NM, 7-Aug-1994.) (Revised by Mario Carneiro, 3-Oct-2016.) (Proof shortened by Wolf Lammen, 6-Mar-2018.)
𝑥𝜑    &   (𝑥 = 𝑦 → (𝜑𝜓))       (𝜑 → ∃𝑥𝜓)

Theoremspimv 2402* A version of spim 2399 with a distinct variable requirement instead of a bound variable hypothesis. See also spimv1 2262 and spimvw 2082. See also spimvALT 2403. (Contributed by NM, 31-Jul-1993.) Removed dependency on ax-10 2168. (Revised by BJ, 29-Nov-2020.)
(𝑥 = 𝑦 → (𝜑𝜓))       (∀𝑥𝜑𝜓)

TheoremspimvALT 2403* Alternate proof of spimv 2402. Shorter but requires more axioms. (Contributed by NM, 31-Jul-1993.) (Proof modification is discouraged.) (New usage is discouraged.)
(𝑥 = 𝑦 → (𝜑𝜓))       (∀𝑥𝜑𝜓)

Theoremspimev 2404* Distinct-variable version of spime 2401. (Contributed by NM, 10-Jan-1993.)
(𝑥 = 𝑦 → (𝜑𝜓))       (𝜑 → ∃𝑥𝜓)

Theoremspv 2405* Specialization, using implicit substitution. (Contributed by NM, 30-Aug-1993.)
(𝑥 = 𝑦 → (𝜑𝜓))       (∀𝑥𝜑𝜓)

Theoremspei 2406 Inference from existential specialization, using implicit substitution. Remove a distinct variable constraint. (Contributed by NM, 19-Aug-1993.) (Proof shortened by Wolf Lammen, 12-May-2018.)
(𝑥 = 𝑦 → (𝜑𝜓))    &   𝜓       𝑥𝜑

Theoremchvar 2407 Implicit substitution of 𝑦 for 𝑥 into a theorem. (Contributed by Raph Levien, 9-Jul-2003.) (Revised by Mario Carneiro, 3-Oct-2016.)
𝑥𝜓    &   (𝑥 = 𝑦 → (𝜑𝜓))    &   𝜑       𝜓

Theoremchvarv 2408* Implicit substitution of 𝑦 for 𝑥 into a theorem. (Contributed by NM, 20-Apr-1994.) (Proof shortened by Wolf Lammen, 22-Apr-2018.) Remove dependency on ax-10 2168. (Revised by Wolf Lammen, 14-Jul-2021.)
(𝑥 = 𝑦 → (𝜑𝜓))    &   𝜑       𝜓

TheoremchvarvOLD 2409* Obsolete proof of chvarv 2408 as of 14-Jul-2021. (Contributed by NM, 20-Apr-1994.) (Proof shortened by Wolf Lammen, 22-Apr-2018.) (Proof modification is discouraged.) (New usage is discouraged.)
(𝑥 = 𝑦 → (𝜑𝜓))    &   𝜑       𝜓

Theoremcbv3 2410 Rule used to change bound variables, using implicit substitution, that does not use ax-c9 34679. (Contributed by NM, 5-Aug-1993.) (Proof shortened by Wolf Lammen, 12-May-2018.)
𝑦𝜑    &   𝑥𝜓    &   (𝑥 = 𝑦 → (𝜑𝜓))       (∀𝑥𝜑 → ∀𝑦𝜓)

Theoremcbv3h 2411 Rule used to change bound variables, using implicit substitution. (Contributed by NM, 8-Jun-1993.) (Proof shortened by Andrew Salmon, 25-May-2011.) (Proof shortened by Wolf Lammen, 12-May-2018.)
(𝜑 → ∀𝑦𝜑)    &   (𝜓 → ∀𝑥𝜓)    &   (𝑥 = 𝑦 → (𝜑𝜓))       (∀𝑥𝜑 → ∀𝑦𝜓)

Theoremcbv1 2412 Rule used to change bound variables, using implicit substitution. (Contributed by NM, 5-Aug-1993.) (Revised by Mario Carneiro, 3-Oct-2016.) Format hypotheses to common style. (Revised by Wolf Lammen, 13-May-2018.)
𝑥𝜑    &   𝑦𝜑    &   (𝜑 → Ⅎ𝑦𝜓)    &   (𝜑 → Ⅎ𝑥𝜒)    &   (𝜑 → (𝑥 = 𝑦 → (𝜓𝜒)))       (𝜑 → (∀𝑥𝜓 → ∀𝑦𝜒))

Theoremcbv1h 2413 Rule used to change bound variables, using implicit substitution. (Contributed by NM, 11-May-1993.) (Proof shortened by Wolf Lammen, 13-May-2018.)
(𝜑 → (𝜓 → ∀𝑦𝜓))    &   (𝜑 → (𝜒 → ∀𝑥𝜒))    &   (𝜑 → (𝑥 = 𝑦 → (𝜓𝜒)))       (∀𝑥𝑦𝜑 → (∀𝑥𝜓 → ∀𝑦𝜒))

Theoremcbv2h 2414 Rule used to change bound variables, using implicit substitution. (Contributed by NM, 11-May-1993.)
(𝜑 → (𝜓 → ∀𝑦𝜓))    &   (𝜑 → (𝜒 → ∀𝑥𝜒))    &   (𝜑 → (𝑥 = 𝑦 → (𝜓𝜒)))       (∀𝑥𝑦𝜑 → (∀𝑥𝜓 ↔ ∀𝑦𝜒))

Theoremcbv2 2415 Rule used to change bound variables, using implicit substitution. (Contributed by NM, 5-Aug-1993.) (Revised by Mario Carneiro, 3-Oct-2016.) Format hypotheses to common style. (Revised by Wolf Lammen, 13-May-2018.)
𝑥𝜑    &   𝑦𝜑    &   (𝜑 → Ⅎ𝑦𝜓)    &   (𝜑 → Ⅎ𝑥𝜒)    &   (𝜑 → (𝑥 = 𝑦 → (𝜓𝜒)))       (𝜑 → (∀𝑥𝜓 ↔ ∀𝑦𝜒))

Theoremcbval 2416 Rule used to change bound variables, using implicit substitution. (Contributed by NM, 13-May-1993.) (Revised by Mario Carneiro, 3-Oct-2016.)
𝑦𝜑    &   𝑥𝜓    &   (𝑥 = 𝑦 → (𝜑𝜓))       (∀𝑥𝜑 ↔ ∀𝑦𝜓)

Theoremcbvex 2417 Rule used to change bound variables, using implicit substitution. (Contributed by NM, 21-Jun-1993.)
𝑦𝜑    &   𝑥𝜓    &   (𝑥 = 𝑦 → (𝜑𝜓))       (∃𝑥𝜑 ↔ ∃𝑦𝜓)

Theoremcbvalv 2418* Rule used to change bound variables, using implicit substitution. (Contributed by NM, 5-Aug-1993.) Remove dependency on ax-10 2168. (Revised by Wolf Lammen, 17-Jul-2021.)
(𝑥 = 𝑦 → (𝜑𝜓))       (∀𝑥𝜑 ↔ ∀𝑦𝜓)

TheoremcbvalvOLD 2419* Obsolete proof of cbvalv 2418 as of 17-Jul-2021. (Contributed by NM, 5-Aug-1993.) (Proof modification is discouraged.) (New usage is discouraged.)
(𝑥 = 𝑦 → (𝜑𝜓))       (∀𝑥𝜑 ↔ ∀𝑦𝜓)

Theoremcbvexv 2420* Rule used to change bound variables, using implicit substitution. (Contributed by NM, 21-Jun-1993.) Remove dependency on ax-10 2168. (Revised by Wolf Lammen, 17-Jul-2021.)
(𝑥 = 𝑦 → (𝜑𝜓))       (∃𝑥𝜑 ↔ ∃𝑦𝜓)

TheoremcbvexvOLD 2421* Obsolete proof of cbvexv 2420 as of 17-Jul-2021. (Contributed by NM, 21-Jun-1993.) (New usage is discouraged.) (Proof modification is discouraged.)
(𝑥 = 𝑦 → (𝜑𝜓))       (∃𝑥𝜑 ↔ ∃𝑦𝜓)

Theoremcbvald 2422* Deduction used to change bound variables, using implicit substitution, particularly useful in conjunction with dvelim 2477. (Contributed by NM, 2-Jan-2002.) (Revised by Mario Carneiro, 6-Oct-2016.) (Revised by Wolf Lammen, 13-May-2018.)
𝑦𝜑    &   (𝜑 → Ⅎ𝑦𝜓)    &   (𝜑 → (𝑥 = 𝑦 → (𝜓𝜒)))       (𝜑 → (∀𝑥𝜓 ↔ ∀𝑦𝜒))

Theoremcbvexd 2423* Deduction used to change bound variables, using implicit substitution, particularly useful in conjunction with dvelim 2477. (Contributed by NM, 2-Jan-2002.) (Revised by Mario Carneiro, 6-Oct-2016.)
𝑦𝜑    &   (𝜑 → Ⅎ𝑦𝜓)    &   (𝜑 → (𝑥 = 𝑦 → (𝜓𝜒)))       (𝜑 → (∃𝑥𝜓 ↔ ∃𝑦𝜒))

Theoremcbval2 2424* Rule used to change bound variables, using implicit substitution. (Contributed by NM, 22-Dec-2003.) (Revised by Mario Carneiro, 6-Oct-2016.) (Proof shortened by Wolf Lammen, 22-Apr-2018.)
𝑧𝜑    &   𝑤𝜑    &   𝑥𝜓    &   𝑦𝜓    &   ((𝑥 = 𝑧𝑦 = 𝑤) → (𝜑𝜓))       (∀𝑥𝑦𝜑 ↔ ∀𝑧𝑤𝜓)

Theoremcbvex2 2425* Rule used to change bound variables, using implicit substitution. (Contributed by NM, 14-Sep-2003.) (Revised by Mario Carneiro, 6-Oct-2016.) (Proof shortened by Wolf Lammen, 16-Jun-2019.)
𝑧𝜑    &   𝑤𝜑    &   𝑥𝜓    &   𝑦𝜓    &   ((𝑥 = 𝑧𝑦 = 𝑤) → (𝜑𝜓))       (∃𝑥𝑦𝜑 ↔ ∃𝑧𝑤𝜓)

Theoremcbvaldva 2426* Rule used to change the bound variable in a universal quantifier with implicit substitution. Deduction form. (Contributed by David Moews, 1-May-2017.) Remove dependency on ax-10 2168. (Revised by Wolf Lammen, 18-Jul-2021.)
((𝜑𝑥 = 𝑦) → (𝜓𝜒))       (𝜑 → (∀𝑥𝜓 ↔ ∀𝑦𝜒))

TheoremcbvaldvaOLD 2427* Obsolete proof of cbvaldva 2426 as of 18-Jul-2021. (Contributed by David Moews, 1-May-2017.) (Proof modification is discouraged.) (New usage is discouraged.)
((𝜑𝑥 = 𝑦) → (𝜓𝜒))       (𝜑 → (∀𝑥𝜓 ↔ ∀𝑦𝜒))

Theoremcbvexdva 2428* Rule used to change the bound variable in an existential quantifier with implicit substitution. Deduction form. (Contributed by David Moews, 1-May-2017.) Remove dependency on ax-10 2168. (Revised by Wolf Lammen, 18-Jul-2021.)
((𝜑𝑥 = 𝑦) → (𝜓𝜒))       (𝜑 → (∃𝑥𝜓 ↔ ∃𝑦𝜒))

TheoremcbvexdvaOLD 2429* Obsolete proof of cbvexdva 2428 as of 18-Jul-2021. (Contributed by David Moews, 1-May-2017.) (Proof modification is discouraged.) (New usage is discouraged.)
((𝜑𝑥 = 𝑦) → (𝜓𝜒))       (𝜑 → (∃𝑥𝜓 ↔ ∃𝑦𝜒))

Theoremcbval2v 2430* Rule used to change bound variables, using implicit substitution. (Contributed by NM, 4-Feb-2005.) Remove dependency on ax-10 2168. (Revised by Wolf Lammen, 18-Jul-2021.)
((𝑥 = 𝑧𝑦 = 𝑤) → (𝜑𝜓))       (∀𝑥𝑦𝜑 ↔ ∀𝑧𝑤𝜓)

Theoremcbval2vOLD 2431* Obsolete proof of cbval2v 2430 as of 18-Jul-2021. (Contributed by NM, 4-Feb-2005.) (Proof modification is discouraged.) (New usage is discouraged.)
((𝑥 = 𝑧𝑦 = 𝑤) → (𝜑𝜓))       (∀𝑥𝑦𝜑 ↔ ∀𝑧𝑤𝜓)

Theoremcbvex2v 2432* Rule used to change bound variables, using implicit substitution. (Contributed by NM, 26-Jul-1995.) Remove dependency on ax-10 2168. (Revised by Wolf Lammen, 18-Jul-2021.)
((𝑥 = 𝑧𝑦 = 𝑤) → (𝜑𝜓))       (∃𝑥𝑦𝜑 ↔ ∃𝑧𝑤𝜓)

Theoremcbvex2vOLD 2433* Obsolete proof of cbvex2v 2432 as of 18-Jul-2021. (Contributed by NM, 26-Jul-1995.) (Proof modification is discouraged.) (New usage is discouraged.)
((𝑥 = 𝑧𝑦 = 𝑤) → (𝜑𝜓))       (∃𝑥𝑦𝜑 ↔ ∃𝑧𝑤𝜓)

Theoremcbvex4v 2434* Rule used to change bound variables, using implicit substitution. (Contributed by NM, 26-Jul-1995.)
((𝑥 = 𝑣𝑦 = 𝑢) → (𝜑𝜓))    &   ((𝑧 = 𝑓𝑤 = 𝑔) → (𝜓𝜒))       (∃𝑥𝑦𝑧𝑤𝜑 ↔ ∃𝑣𝑢𝑓𝑔𝜒)

Theoremequs4 2435 Lemma used in proofs of implicit substitution properties. The converse requires either a dv condition (sb56 2297) or a non-freeness hypothesis (equs45f 2487). See equs4v 2085 for a version requiring fewer axioms. (Contributed by NM, 10-May-1993.) (Proof shortened by Mario Carneiro, 20-May-2014.) (Proof shortened by Wolf Lammen, 5-Feb-2018.)
(∀𝑥(𝑥 = 𝑦𝜑) → ∃𝑥(𝑥 = 𝑦𝜑))

Theoremequsal 2436 An equivalence related to implicit substitution. See equsalvw 2086 and equsalv 2255 for versions with dv conditions proved from fewer axioms. See also the dual form equsex 2437. (Contributed by NM, 2-Jun-1993.) (Proof shortened by Andrew Salmon, 12-Aug-2011.) (Revised by Mario Carneiro, 3-Oct-2016.) (Proof shortened by Wolf Lammen, 5-Feb-2018.)
𝑥𝜓    &   (𝑥 = 𝑦 → (𝜑𝜓))       (∀𝑥(𝑥 = 𝑦𝜑) ↔ 𝜓)

Theoremequsex 2437 An equivalence related to implicit substitution. See equsexvw 2087 and equsexv 2256 for versions with dv conditions proved from fewer axioms. See also the dual form equsal 2436. See equsexALT 2438 for an alternate proof. (Contributed by NM, 5-Aug-1993.) (Revised by Mario Carneiro, 3-Oct-2016.) (Proof shortened by Wolf Lammen, 6-Feb-2018.)
𝑥𝜓    &   (𝑥 = 𝑦 → (𝜑𝜓))       (∃𝑥(𝑥 = 𝑦𝜑) ↔ 𝜓)

TheoremequsexALT 2438 Alternate proof of equsex 2437. This proves the result directly, instead of as a corollary of equsal 2436 via equs4 2435. Note in particular that only existential quantifiers appear in the proof and that the only step requiring ax-13 2391 is ax6e 2395. This proof mimics that of equsal 2436 (in particular, note that pm5.32i 672, exbii 1923, 19.41 2250, mpbiran 991 correspond respectively to pm5.74i 260, albii 1896, 19.23 2227, a1bi 351). (Contributed by BJ, 20-Aug-2020.) (Proof modification is discouraged.) (New usage is discouraged.)
𝑥𝜓    &   (𝑥 = 𝑦 → (𝜑𝜓))       (∃𝑥(𝑥 = 𝑦𝜑) ↔ 𝜓)

Theoremequsalh 2439 An equivalence related to implicit substitution. See equsalhw 2270 for a version with a dv condition requiring fewer axioms. (Contributed by NM, 2-Jun-1993.)
(𝜓 → ∀𝑥𝜓)    &   (𝑥 = 𝑦 → (𝜑𝜓))       (∀𝑥(𝑥 = 𝑦𝜑) ↔ 𝜓)

Theoremequsexh 2440 An equivalence related to implicit substitution. See equsexhv 2271 for a version with a dv condition which does not require ax-13 2391. (Contributed by NM, 5-Aug-1993.)
(𝜓 → ∀𝑥𝜓)    &   (𝑥 = 𝑦 → (𝜑𝜓))       (∃𝑥(𝑥 = 𝑦𝜑) ↔ 𝜓)

Theoremax13lem2 2441* Lemma for nfeqf2 2442. This lemma is equivalent to ax13v 2392 with one distinct variable constraint removed. (Contributed by Wolf Lammen, 8-Sep-2018.) Reduce axiom usage. (Revised by Wolf Lammen, 18-Oct-2020.) (New usage is discouraged.)
𝑥 = 𝑦 → (∃𝑥 𝑧 = 𝑦𝑧 = 𝑦))

Theoremnfeqf2 2442* An equation between setvar is free of any other setvar. (Contributed by Wolf Lammen, 9-Jun-2019.)
(¬ ∀𝑥 𝑥 = 𝑦 → Ⅎ𝑥 𝑧 = 𝑦)

Theoremdveeq2 2443* Quantifier introduction when one pair of variables is distinct. (Contributed by NM, 2-Jan-2002.) (Revised by NM, 20-Jul-2015.) Remove dependency on ax-11 2183. (Revised by Wolf Lammen, 8-Sep-2018.)
(¬ ∀𝑥 𝑥 = 𝑦 → (𝑧 = 𝑦 → ∀𝑥 𝑧 = 𝑦))

Theoremnfeqf1 2444* An equation between setvar is free of any other setvar. (Contributed by Wolf Lammen, 10-Jun-2019.)
(¬ ∀𝑥 𝑥 = 𝑦 → Ⅎ𝑥 𝑦 = 𝑧)

Theoremdveeq1 2445* Quantifier introduction when one pair of variables is distinct. (Contributed by NM, 2-Jan-2002.) Remove dependency on ax-11 2183. (Revised by Wolf Lammen, 8-Sep-2018.)
(¬ ∀𝑥 𝑥 = 𝑦 → (𝑦 = 𝑧 → ∀𝑥 𝑦 = 𝑧))

Theoremnfeqf 2446 A variable is effectively not free in an equality if it is not either of the involved variables. version of ax-c9 34679. (Contributed by Mario Carneiro, 6-Oct-2016.) Remove dependency on ax-11 2183. (Revised by Wolf Lammen, 6-Sep-2018.)
((¬ ∀𝑧 𝑧 = 𝑥 ∧ ¬ ∀𝑧 𝑧 = 𝑦) → Ⅎ𝑧 𝑥 = 𝑦)

Theoremaxc9 2447 Derive set.mm's original ax-c9 34679 from the shorter ax-13 2391. (Contributed by NM, 29-Nov-2015.) (Revised by NM, 24-Dec-2015.) (Proof shortened by Wolf Lammen, 29-Apr-2018.)
(¬ ∀𝑧 𝑧 = 𝑥 → (¬ ∀𝑧 𝑧 = 𝑦 → (𝑥 = 𝑦 → ∀𝑧 𝑥 = 𝑦)))

Theoremaxc15 2448 Derivation of set.mm's original ax-c15 34678 from ax-c11n 34677 and the shorter ax-12 2196 that has replaced it.

Theorem ax12 2449 shows the reverse derivation of ax-12 2196 from ax-c15 34678.

Normally, axc15 2448 should be used rather than ax-c15 34678, except by theorems specifically studying the latter's properties. (Contributed by NM, 2-Feb-2007.) (Proof shortened by Wolf Lammen, 21-Apr-2018.)

(¬ ∀𝑥 𝑥 = 𝑦 → (𝑥 = 𝑦 → (𝜑 → ∀𝑥(𝑥 = 𝑦𝜑))))

Theoremax12 2449 Rederivation of axiom ax-12 2196 from ax12v 2197 (used only via sp 2200) , axc11r 2332, and axc15 2448 (on top of Tarski's FOL). (Contributed by NM, 22-Jan-2007.) Proof uses contemporary axioms. (Revised by Wolf Lammen, 8-Aug-2020.) (Proof shortened by BJ, 4-Jul-2021.)
(𝑥 = 𝑦 → (∀𝑦𝜑 → ∀𝑥(𝑥 = 𝑦𝜑)))

Theoremax13ALT 2450 Alternate proof of ax13 2394 from FOL, sp 2200, and axc9 2447. (Contributed by NM, 21-Dec-2015.) (Proof shortened by Wolf Lammen, 31-Jan-2018.) (Proof modification is discouraged.) (New usage is discouraged.)
𝑥 = 𝑦 → (𝑦 = 𝑧 → ∀𝑥 𝑦 = 𝑧))

Theoremaxc11n 2451 Derive set.mm's original ax-c11n 34677 from others. Commutation law for identical variable specifiers. The antecedent and consequent are true when 𝑥 and 𝑦 are substituted with the same variable. Lemma L12 in [Megill] p. 445 (p. 12 of the preprint). If a disjoint variable condition is added on 𝑥 and 𝑦, then this becomes an instance of aevlem 2132. Use aecom 2453 instead when this does not lengthen the proof. (Contributed by NM, 10-May-1993.) (Revised by NM, 7-Nov-2015.) (Proof shortened by Wolf Lammen, 6-Mar-2018.) (Revised by Wolf Lammen, 30-Nov-2019.) (Proof shortened by BJ, 29-Mar-2021.) (Proof shortened by Wolf Lammen, 2-Jul-2021.)
(∀𝑥 𝑥 = 𝑦 → ∀𝑦 𝑦 = 𝑥)

Theoremaxc11nOLD 2452 Obsolete proof of axc11n 2451 as of 2-Jul-2021. (Contributed by NM, 10-May-1993.) (Revised by NM, 7-Nov-2015.) (Proof shortened by Wolf Lammen, 6-Mar-2018.) (Revised by Wolf Lammen, 30-Nov-2019.) (Proof shortened by BJ, 29-Mar-2021.) (Proof modification is discouraged.) (New usage is discouraged.)
(∀𝑥 𝑥 = 𝑦 → ∀𝑦 𝑦 = 𝑥)

Theoremaecom 2453 Commutation law for identical variable specifiers. Both sides of the biconditional are true when 𝑥 and 𝑦 are substituted with the same variable. (Contributed by NM, 10-May-1993.) Changed to a biconditional. (Revised by BJ, 26-Sep-2019.)
(∀𝑥 𝑥 = 𝑦 ↔ ∀𝑦 𝑦 = 𝑥)

Theoremaecoms 2454 A commutation rule for identical variable specifiers. (Contributed by NM, 10-May-1993.)
(∀𝑥 𝑥 = 𝑦𝜑)       (∀𝑦 𝑦 = 𝑥𝜑)

Theoremnaecoms 2455 A commutation rule for distinct variable specifiers. (Contributed by NM, 2-Jan-2002.)
(¬ ∀𝑥 𝑥 = 𝑦𝜑)       (¬ ∀𝑦 𝑦 = 𝑥𝜑)

Theoremaxc11 2456 Show that ax-c11 34676 can be derived from ax-c11n 34677 in the form of axc11n 2451. Normally, axc11 2456 should be used rather than ax-c11 34676, except by theorems specifically studying the latter's properties. (Contributed by NM, 16-May-2008.) (Proof shortened by Wolf Lammen, 21-Apr-2018.)
(∀𝑥 𝑥 = 𝑦 → (∀𝑥𝜑 → ∀𝑦𝜑))

Theoremhbae 2457 All variables are effectively bound in an identical variable specifier. (Contributed by NM, 13-May-1993.) (Proof shortened by Wolf Lammen, 21-Apr-2018.)
(∀𝑥 𝑥 = 𝑦 → ∀𝑧𝑥 𝑥 = 𝑦)

Theoremnfae 2458 All variables are effectively bound in an identical variable specifier. (Contributed by Mario Carneiro, 11-Aug-2016.)
𝑧𝑥 𝑥 = 𝑦

Theoremhbnae 2459 All variables are effectively bound in a distinct variable specifier. Lemma L19 in [Megill] p. 446 (p. 14 of the preprint). (Contributed by NM, 13-May-1993.)
(¬ ∀𝑥 𝑥 = 𝑦 → ∀𝑧 ¬ ∀𝑥 𝑥 = 𝑦)

Theoremnfnae 2460 All variables are effectively bound in a distinct variable specifier. (Contributed by Mario Carneiro, 11-Aug-2016.)
𝑧 ¬ ∀𝑥 𝑥 = 𝑦

Theoremhbnaes 2461 Rule that applies hbnae 2459 to antecedent. (Contributed by NM, 15-May-1993.)
(∀𝑧 ¬ ∀𝑥 𝑥 = 𝑦𝜑)       (¬ ∀𝑥 𝑥 = 𝑦𝜑)

Theoremaxc16i 2462* Inference with axc16 2282 as its conclusion. (Contributed by NM, 20-May-2008.) (Proof modification is discouraged.)
(𝑥 = 𝑧 → (𝜑𝜓))    &   (𝜓 → ∀𝑥𝜓)       (∀𝑥 𝑥 = 𝑦 → (𝜑 → ∀𝑥𝜑))

Theoremaxc16nfALT 2463* Alternate proof of axc16nf 2284, shorter but requiring ax-11 2183 and ax-13 2391. (Contributed by Mario Carneiro, 7-Oct-2016.) (Proof modification is discouraged.) (New usage is discouraged.)
(∀𝑥 𝑥 = 𝑦 → Ⅎ𝑧𝜑)

Theoremdral2 2464 Formula-building lemma for use with the Distinctor Reduction Theorem. Part of Theorem 9.4 of [Megill] p. 448 (p. 16 of preprint). (Contributed by NM, 27-Feb-2005.) Allow a shortening of dral1 2465. (Revised by Wolf Lammen, 4-Mar-2018.)
(∀𝑥 𝑥 = 𝑦 → (𝜑𝜓))       (∀𝑥 𝑥 = 𝑦 → (∀𝑧𝜑 ↔ ∀𝑧𝜓))

Theoremdral1 2465 Formula-building lemma for use with the Distinctor Reduction Theorem. Part of Theorem 9.4 of [Megill] p. 448 (p. 16 of preprint). (Contributed by NM, 24-Nov-1994.) Remove dependency on ax-11 2183. (Revised by Wolf Lammen, 6-Sep-2018.)
(∀𝑥 𝑥 = 𝑦 → (𝜑𝜓))       (∀𝑥 𝑥 = 𝑦 → (∀𝑥𝜑 ↔ ∀𝑦𝜓))

Theoremdral1ALT 2466 Alternate proof of dral1 2465, shorter but requiring ax-11 2183. (Contributed by NM, 24-Nov-1994.) (Proof shortened by Wolf Lammen, 22-Apr-2018.) (New usage is discouraged.) (Proof modification is discouraged.)
(∀𝑥 𝑥 = 𝑦 → (𝜑𝜓))       (∀𝑥 𝑥 = 𝑦 → (∀𝑥𝜑 ↔ ∀𝑦𝜓))

Theoremdrex1 2467 Formula-building lemma for use with the Distinctor Reduction Theorem. Part of Theorem 9.4 of [Megill] p. 448 (p. 16 of preprint). (Contributed by NM, 27-Feb-2005.)
(∀𝑥 𝑥 = 𝑦 → (𝜑𝜓))       (∀𝑥 𝑥 = 𝑦 → (∃𝑥𝜑 ↔ ∃𝑦𝜓))

Theoremdrex2 2468 Formula-building lemma for use with the Distinctor Reduction Theorem. Part of Theorem 9.4 of [Megill] p. 448 (p. 16 of preprint). (Contributed by NM, 27-Feb-2005.)
(∀𝑥 𝑥 = 𝑦 → (𝜑𝜓))       (∀𝑥 𝑥 = 𝑦 → (∃𝑧𝜑 ↔ ∃𝑧𝜓))

Theoremdrnf1 2469 Formula-building lemma for use with the Distinctor Reduction Theorem. (Contributed by Mario Carneiro, 4-Oct-2016.)
(∀𝑥 𝑥 = 𝑦 → (𝜑𝜓))       (∀𝑥 𝑥 = 𝑦 → (Ⅎ𝑥𝜑 ↔ Ⅎ𝑦𝜓))

Theoremdrnf2 2470 Formula-building lemma for use with the Distinctor Reduction Theorem. (Contributed by Mario Carneiro, 4-Oct-2016.) (Proof shortened by Wolf Lammen, 5-May-2018.)
(∀𝑥 𝑥 = 𝑦 → (𝜑𝜓))       (∀𝑥 𝑥 = 𝑦 → (Ⅎ𝑧𝜑 ↔ Ⅎ𝑧𝜓))

Theoremnfald2 2471 Variation on nfald 2310 which adds the hypothesis that 𝑥 and 𝑦 are distinct in the inner subproof. (Contributed by Mario Carneiro, 8-Oct-2016.)
𝑦𝜑    &   ((𝜑 ∧ ¬ ∀𝑥 𝑥 = 𝑦) → Ⅎ𝑥𝜓)       (𝜑 → Ⅎ𝑥𝑦𝜓)

Theoremnfexd2 2472 Variation on nfexd 2312 which adds the hypothesis that 𝑥 and 𝑦 are distinct in the inner subproof. (Contributed by Mario Carneiro, 8-Oct-2016.)
𝑦𝜑    &   ((𝜑 ∧ ¬ ∀𝑥 𝑥 = 𝑦) → Ⅎ𝑥𝜓)       (𝜑 → Ⅎ𝑥𝑦𝜓)

Theoremexdistrf 2473 Distribution of existential quantifiers, with a bound-variable hypothesis saying that 𝑦 is not free in 𝜑, but 𝑥 can be free in 𝜑 (and there is no distinct variable condition on 𝑥 and 𝑦). (Contributed by Mario Carneiro, 20-Mar-2013.) (Proof shortened by Wolf Lammen, 14-May-2018.)
(¬ ∀𝑥 𝑥 = 𝑦 → Ⅎ𝑦𝜑)       (∃𝑥𝑦(𝜑𝜓) → ∃𝑥(𝜑 ∧ ∃𝑦𝜓))

Theoremdvelimf 2474 Version of dvelimv 2478 without any variable restrictions. (Contributed by NM, 1-Oct-2002.) (Revised by Mario Carneiro, 6-Oct-2016.) (Proof shortened by Wolf Lammen, 11-May-2018.)
𝑥𝜑    &   𝑧𝜓    &   (𝑧 = 𝑦 → (𝜑𝜓))       (¬ ∀𝑥 𝑥 = 𝑦 → Ⅎ𝑥𝜓)

Theoremdvelimdf 2475 Deduction form of dvelimf 2474. (Contributed by NM, 7-Apr-2004.) (Revised by Mario Carneiro, 6-Oct-2016.) (Proof shortened by Wolf Lammen, 11-May-2018.)
𝑥𝜑    &   𝑧𝜑    &   (𝜑 → Ⅎ𝑥𝜓)    &   (𝜑 → Ⅎ𝑧𝜒)    &   (𝜑 → (𝑧 = 𝑦 → (𝜓𝜒)))       (𝜑 → (¬ ∀𝑥 𝑥 = 𝑦 → Ⅎ𝑥𝜒))

Theoremdvelimh 2476 Version of dvelim 2477 without any variable restrictions. (Contributed by NM, 1-Oct-2002.) (Proof shortened by Wolf Lammen, 11-May-2018.)
(𝜑 → ∀𝑥𝜑)    &   (𝜓 → ∀𝑧𝜓)    &   (𝑧 = 𝑦 → (𝜑𝜓))       (¬ ∀𝑥 𝑥 = 𝑦 → (𝜓 → ∀𝑥𝜓))

Theoremdvelim 2477* This theorem can be used to eliminate a distinct variable restriction on 𝑥 and 𝑧 and replace it with the "distinctor" ¬ ∀𝑥𝑥 = 𝑦 as an antecedent. 𝜑 normally has 𝑧 free and can be read 𝜑(𝑧), and 𝜓 substitutes 𝑦 for 𝑧 and can be read 𝜑(𝑦). We do not require that 𝑥 and 𝑦 be distinct: if they are not, the distinctor will become false (in multiple-element domains of discourse) and "protect" the consequent.

To obtain a closed-theorem form of this inference, prefix the hypotheses with 𝑥𝑧, conjoin them, and apply dvelimdf 2475.

Other variants of this theorem are dvelimh 2476 (with no distinct variable restrictions) and dvelimhw 2318 (that avoids ax-13 2391). (Contributed by NM, 23-Nov-1994.)

(𝜑 → ∀𝑥𝜑)    &   (𝑧 = 𝑦 → (𝜑𝜓))       (¬ ∀𝑥 𝑥 = 𝑦 → (𝜓 → ∀𝑥𝜓))

Theoremdvelimv 2478* Similar to dvelim 2477 with first hypothesis replaced by a distinct variable condition. (Contributed by NM, 25-Jul-2015.) (Proof shortened by Wolf Lammen, 30-Apr-2018.)
(𝑧 = 𝑦 → (𝜑𝜓))       (¬ ∀𝑥 𝑥 = 𝑦 → (𝜓 → ∀𝑥𝜓))

Theoremdvelimnf 2479* Version of dvelim 2477 using "not free" notation. (Contributed by Mario Carneiro, 9-Oct-2016.)
𝑥𝜑    &   (𝑧 = 𝑦 → (𝜑𝜓))       (¬ ∀𝑥 𝑥 = 𝑦 → Ⅎ𝑥𝜓)

Theoremdveeq2ALT 2480* Alternate proof of dveeq2 2443, shorter but requiring ax-11 2183. (Contributed by NM, 2-Jan-2002.) (Revised by NM, 20-Jul-2015.) (New usage is discouraged.) (Proof modification is discouraged.)
(¬ ∀𝑥 𝑥 = 𝑦 → (𝑧 = 𝑦 → ∀𝑥 𝑧 = 𝑦))

Theoremax12OLD 2481 Obsolete proof of ax12 2449 as of 4-Jul-2021 . Rederivation of axiom ax-12 2196 from ax12v 2197, axc11r 2332, and other axioms. (Contributed by NM, 22-Jan-2007.) Proof uses contemporary axioms. (Revised by Wolf Lammen, 8-Aug-2020.) (Proof modification is discouraged.) (New usage is discouraged.)
(𝑥 = 𝑦 → (∀𝑦𝜑 → ∀𝑥(𝑥 = 𝑦𝜑)))

Theoremax12b 2482 A bidirectional version of axc15 2448. (Contributed by NM, 30-Jun-2006.)
((¬ ∀𝑥 𝑥 = 𝑦𝑥 = 𝑦) → (𝜑 ↔ ∀𝑥(𝑥 = 𝑦𝜑)))

Theoremequvini 2483 A variable introduction law for equality. Lemma 15 of [Monk2] p. 109, however we do not require 𝑧 to be distinct from 𝑥 and 𝑦. See equvinv 2112 for a shorter proof requiring fewer axioms when 𝑧 is required to be distinct from 𝑥 and 𝑦. (Contributed by NM, 10-Jan-1993.) (Proof shortened by Andrew Salmon, 25-May-2011.) (Proof shortened by Wolf Lammen, 15-Sep-2018.)
(𝑥 = 𝑦 → ∃𝑧(𝑥 = 𝑧𝑧 = 𝑦))

Theoremequvel 2484 A variable elimination law for equality with no distinct variable requirements. Compare equvini 2483. (Contributed by NM, 1-Mar-2013.) (Proof shortened by Mario Carneiro, 17-Oct-2016.) (Proof shortened by Wolf Lammen, 15-Jun-2019.)
(∀𝑧(𝑧 = 𝑥𝑧 = 𝑦) → 𝑥 = 𝑦)

Theoremequs5a 2485 A property related to substitution that unlike equs5 2488 does not require a distinctor antecedent. See equs5aALT 2322 for an alternate proof using ax-12 2196 but not ax13 2394. (Contributed by NM, 2-Feb-2007.)
(∃𝑥(𝑥 = 𝑦 ∧ ∀𝑦𝜑) → ∀𝑥(𝑥 = 𝑦𝜑))

Theoremequs5e 2486 A property related to substitution that unlike equs5 2488 does not require a distinctor antecedent. See equs5eALT 2323 for an alternate proof using ax-12 2196 but not ax13 2394. (Contributed by NM, 2-Feb-2007.) (Proof shortened by Wolf Lammen, 15-Jan-2018.)
(∃𝑥(𝑥 = 𝑦𝜑) → ∀𝑥(𝑥 = 𝑦 → ∃𝑦𝜑))

Theoremequs45f 2487 Two ways of expressing substitution when 𝑦 is not free in 𝜑. The implication "to the left" is equs4 2435 and does not require the non-freeness hypothesis. Theorem sb56 2297 replaces the non-freeness hypothesis with a dv condition and equs5 2488 replaces it with a distinctor as antecedent. (Contributed by NM, 25-Apr-2008.) (Revised by Mario Carneiro, 4-Oct-2016.)
𝑦𝜑       (∃𝑥(𝑥 = 𝑦𝜑) ↔ ∀𝑥(𝑥 = 𝑦𝜑))

Theoremequs5 2488 Lemma used in proofs of substitution properties. If there is a dv condition on 𝑥, 𝑦, then sb56 2297 can be used instead; if 𝑦 is not free in 𝜑, then equs45f 2487 can be used. (Contributed by NM, 14-May-1993.) (Revised by BJ, 1-Oct-2018.)
(¬ ∀𝑥 𝑥 = 𝑦 → (∃𝑥(𝑥 = 𝑦𝜑) ↔ ∀𝑥(𝑥 = 𝑦𝜑)))

Theoremsb2 2489 One direction of a simplified definition of substitution. The converse requires either a dv condition (sb6 2566) or a non-freeness hypothesis (sb6f 2522). (Contributed by NM, 13-May-1993.)
(∀𝑥(𝑥 = 𝑦𝜑) → [𝑦 / 𝑥]𝜑)

Theoremstdpc4 2490 The specialization axiom of standard predicate calculus. It states that if a statement 𝜑 holds for all 𝑥, then it also holds for the specific case of 𝑦 (properly) substituted for 𝑥. Translated to traditional notation, it can be read: "𝑥𝜑(𝑥) → 𝜑(𝑦), provided that 𝑦 is free for 𝑥 in 𝜑(𝑥)." Axiom 4 of [Mendelson] p. 69. See also spsbc 3589 and rspsbc 3659. (Contributed by NM, 14-May-1993.)
(∀𝑥𝜑 → [𝑦 / 𝑥]𝜑)

Theorem2stdpc4 2491 A double specialization using explicit substitution. This is Theorem PM*11.1 in [WhiteheadRussell] p. 159. See stdpc4 2490 for the analogous single specialization. See 2sp 2203 for another double specialization. (Contributed by Andrew Salmon, 24-May-2011.) (Revised by BJ, 21-Oct-2018.)
(∀𝑥𝑦𝜑 → [𝑧 / 𝑥][𝑤 / 𝑦]𝜑)

Theoremsb3 2492 One direction of a simplified definition of substitution when variables are distinct. (Contributed by NM, 5-Aug-1993.)
(¬ ∀𝑥 𝑥 = 𝑦 → (∃𝑥(𝑥 = 𝑦𝜑) → [𝑦 / 𝑥]𝜑))

Theoremsb4 2493 One direction of a simplified definition of substitution when variables are distinct. (Contributed by NM, 14-May-1993.)
(¬ ∀𝑥 𝑥 = 𝑦 → ([𝑦 / 𝑥]𝜑 → ∀𝑥(𝑥 = 𝑦𝜑)))

Theoremsb4a 2494 A version of sb4 2493 that doesn't require a distinctor antecedent. (Contributed by NM, 2-Feb-2007.)
([𝑦 / 𝑥]∀𝑦𝜑 → ∀𝑥(𝑥 = 𝑦𝜑))

Theoremsb4b 2495 Simplified definition of substitution when variables are distinct. (Contributed by NM, 27-May-1997.)
(¬ ∀𝑥 𝑥 = 𝑦 → ([𝑦 / 𝑥]𝜑 ↔ ∀𝑥(𝑥 = 𝑦𝜑)))

Theoremhbsb2 2496 Bound-variable hypothesis builder for substitution. (Contributed by NM, 14-May-1993.)
(¬ ∀𝑥 𝑥 = 𝑦 → ([𝑦 / 𝑥]𝜑 → ∀𝑥[𝑦 / 𝑥]𝜑))

Theoremnfsb2 2497 Bound-variable hypothesis builder for substitution. (Contributed by Mario Carneiro, 4-Oct-2016.)
(¬ ∀𝑥 𝑥 = 𝑦 → Ⅎ𝑥[𝑦 / 𝑥]𝜑)

Theoremhbsb2a 2498 Special case of a bound-variable hypothesis builder for substitution. (Contributed by NM, 2-Feb-2007.)
([𝑦 / 𝑥]∀𝑦𝜑 → ∀𝑥[𝑦 / 𝑥]𝜑)

Theoremsb4e 2499 One direction of a simplified definition of substitution that unlike sb4 2493 doesn't require a distinctor antecedent. (Contributed by NM, 2-Feb-2007.)
([𝑦 / 𝑥]𝜑 → ∀𝑥(𝑥 = 𝑦 → ∃𝑦𝜑))

Theoremhbsb2e 2500 Special case of a bound-variable hypothesis builder for substitution. (Contributed by NM, 2-Feb-2007.)
([𝑦 / 𝑥]𝜑 → ∀𝑥[𝑦 / 𝑥]∃𝑦𝜑)

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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 42801-42900 430 42901-43000 431 43001-43064
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