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

Theoremspim 2401 Specialization, using implicit substitution. Compare Lemma 14 of [Tarski] p. 70. The spim 2401 series of theorems requires that only one direction of the substitution hypothesis hold. Usage of this theorem is discouraged because it depends on ax-13 2386. Check out spimw 1969 for a version requiring less axioms. (Contributed by NM, 10-Jan-1993.) (Revised by Mario Carneiro, 3-Oct-2016.) (Proof shortened by Wolf Lammen, 18-Feb-2018.) (New usage is discouraged.)
𝑥𝜓    &   (𝑥 = 𝑦 → (𝜑𝜓))       (∀𝑥𝜑𝜓)

Theoremspimed 2402 Deduction version of spime 2403. Usage of this theorem is discouraged because it depends on ax-13 2386. Use the weaker spimedv 2192 if possible. (Contributed by NM, 14-May-1993.) (Revised by Mario Carneiro, 3-Oct-2016.) (Proof shortened by Wolf Lammen, 19-Feb-2018.) (New usage is discouraged.)
(𝜒 → Ⅎ𝑥𝜑)    &   (𝑥 = 𝑦 → (𝜑𝜓))       (𝜒 → (𝜑 → ∃𝑥𝜓))

Theoremspime 2403 Existential introduction, using implicit substitution. Compare Lemma 14 of [Tarski] p. 70. Usage of this theorem is discouraged because it depends on ax-13 2386. Check out spimew 1970 for a weaker version requiring less axioms. (Contributed by NM, 7-Aug-1994.) (Revised by Mario Carneiro, 3-Oct-2016.) (Proof shortened by Wolf Lammen, 6-Mar-2018.) (New usage is discouraged.)
𝑥𝜑    &   (𝑥 = 𝑦 → (𝜑𝜓))       (𝜑 → ∃𝑥𝜓)

Theoremspimv 2404* A version of spim 2401 with a distinct variable requirement instead of a bound-variable hypothesis. Usage of this theorem is discouraged because it depends on ax-13 2386. See spimfv 2236 and spimvw 1998 for versions requiring fewer axioms. (Contributed by NM, 31-Jul-1993.) (New usage is discouraged.)
(𝑥 = 𝑦 → (𝜑𝜓))       (∀𝑥𝜑𝜓)

TheoremspimvALT 2405* Alternate proof of spimv 2404. Note that it requires only ax-1 6 through ax-5 1907 together with ax6e 2397. Currently, proofs derive from ax6v 1967, but if ax-6 1966 could be used instead, this proof would reduce axiom usage. (Contributed by NM, 31-Jul-1993.) Remove dependency on ax-10 2141. (Revised by BJ, 29-Nov-2020.) (Proof modification is discouraged.) (New usage is discouraged.)
(𝑥 = 𝑦 → (𝜑𝜓))       (∀𝑥𝜑𝜓)

Theoremspimev 2406* Distinct-variable version of spime 2403. Usage of this theorem is discouraged because it depends on ax-13 2386. Use the weaker spimevw 1997 if possible. (Contributed by NM, 10-Jan-1993.) (New usage is discouraged.)
(𝑥 = 𝑦 → (𝜑𝜓))       (𝜑 → ∃𝑥𝜓)

Theoremspv 2407* Specialization, using implicit substitution. Usage of this theorem is discouraged because it depends on ax-13 2386. Use the weaker spvv 1999 if possible. (Contributed by NM, 30-Aug-1993.) (New usage is discouraged.)
(𝑥 = 𝑦 → (𝜑𝜓))       (∀𝑥𝜑𝜓)

Theoremspei 2408 Inference from existential specialization, using implicit substitution. Remove a distinct variable constraint. Usage of this theorem is discouraged because it depends on ax-13 2386. Use the weaker speiv 1972 if possible. (Contributed by NM, 19-Aug-1993.) (Proof shortened by Wolf Lammen, 12-May-2018.) (New usage is discouraged.)
(𝑥 = 𝑦 → (𝜑𝜓))    &   𝜓       𝑥𝜑

Theoremchvar 2409 Implicit substitution of 𝑦 for 𝑥 into a theorem. Usage of this theorem is discouraged because it depends on ax-13 2386. Use the weaker chvarfv 2237 if possible. (Contributed by Raph Levien, 9-Jul-2003.) (Revised by Mario Carneiro, 3-Oct-2016.) (New usage is discouraged.)
𝑥𝜓    &   (𝑥 = 𝑦 → (𝜑𝜓))    &   𝜑       𝜓

Theoremchvarv 2410* Implicit substitution of 𝑦 for 𝑥 into a theorem. Usage of this theorem is discouraged because it depends on ax-13 2386. Use the weaker chvarvv 2001 if possible. (Contributed by NM, 20-Apr-1994.) (Proof shortened by Wolf Lammen, 22-Apr-2018.) (New usage is discouraged.)
(𝑥 = 𝑦 → (𝜑𝜓))    &   𝜑       𝜓

Theoremcbv3 2411 Rule used to change bound variables, using implicit substitution, that does not use ax-c9 36020. Usage of this theorem is discouraged because it depends on ax-13 2386. Use the weaker cbv3v 2351 if possible. (Contributed by NM, 5-Aug-1993.) (Proof shortened by Wolf Lammen, 12-May-2018.) (New usage is discouraged.)
𝑦𝜑    &   𝑥𝜓    &   (𝑥 = 𝑦 → (𝜑𝜓))       (∀𝑥𝜑 → ∀𝑦𝜓)

Theoremcbval 2412 Rule used to change bound variables, using implicit substitution. Usage of this theorem is discouraged because it depends on ax-13 2386. Check out cbvalw 2038, cbvalvw 2039, cbvalv1 2357 for versions requiring fewer axioms. (Contributed by NM, 13-May-1993.) (Revised by Mario Carneiro, 3-Oct-2016.) (New usage is discouraged.)
𝑦𝜑    &   𝑥𝜓    &   (𝑥 = 𝑦 → (𝜑𝜓))       (∀𝑥𝜑 ↔ ∀𝑦𝜓)

Theoremcbvex 2413 Rule used to change bound variables, using implicit substitution. Usage of this theorem is discouraged because it depends on ax-13 2386. Check out cbvexvw 2040, cbvexv1 2358 for weaker versions requiring fewer axioms. (Contributed by NM, 21-Jun-1993.) (New usage is discouraged.)
𝑦𝜑    &   𝑥𝜓    &   (𝑥 = 𝑦 → (𝜑𝜓))       (∃𝑥𝜑 ↔ ∃𝑦𝜓)

Theoremcbvalv 2414* Rule used to change bound variables, using implicit substitution. Usage of this theorem is discouraged because it depends on ax-13 2386. See cbvalvw 2039 for a version requiring fewer axioms, to be preferred when sufficient. (Contributed by NM, 5-Aug-1993.) Remove dependency on ax-10 2141, shorten. (Revised by Wolf Lammen, 11-Sep-2023.) (New usage is discouraged.)
(𝑥 = 𝑦 → (𝜑𝜓))       (∀𝑥𝜑 ↔ ∀𝑦𝜓)

Theoremcbvexv 2415* Rule used to change bound variables, using implicit substitution. Usage of this theorem is discouraged because it depends on ax-13 2386. See cbvexvw 2040 for a version requiring fewer axioms, to be preferred when sufficient. (Contributed by NM, 21-Jun-1993.) Remove dependency on ax-10 2141, shorten. (Revised by Wolf Lammen, 11-Sep-2023.) (New usage is discouraged.)
(𝑥 = 𝑦 → (𝜑𝜓))       (∃𝑥𝜑 ↔ ∃𝑦𝜓)

TheoremcbvalvOLD 2416* Obsolete version of cbvalv 2414 as of 11-Sep-2023. (Contributed by NM, 5-Aug-1993.) Remove dependency on ax-10 2141. (Revised by Wolf Lammen, 17-Jul-2021.) (Proof modification is discouraged.) (New usage is discouraged.)
(𝑥 = 𝑦 → (𝜑𝜓))       (∀𝑥𝜑 ↔ ∀𝑦𝜓)

TheoremcbvexvOLD 2417* Obsolete version of cbvexv 2415 as of 11-Sep-2023. (Contributed by NM, 21-Jun-1993.) Remove dependency on ax-10 2141. (Revised by Wolf Lammen, 17-Jul-2021.) (Proof modification is discouraged.) (New usage is discouraged.)
(𝑥 = 𝑦 → (𝜑𝜓))       (∃𝑥𝜑 ↔ ∃𝑦𝜓)

Theoremcbv1 2418 Rule used to change bound variables, using implicit substitution. Usage of this theorem is discouraged because it depends on ax-13 2386. See cbv1v 2352 with disjoint variable conditions, not depending on ax-13 2386. (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.) (New usage is discouraged.)
𝑥𝜑    &   𝑦𝜑    &   (𝜑 → Ⅎ𝑦𝜓)    &   (𝜑 → Ⅎ𝑥𝜒)    &   (𝜑 → (𝑥 = 𝑦 → (𝜓𝜒)))       (𝜑 → (∀𝑥𝜓 → ∀𝑦𝜒))

Theoremcbv2 2419 Rule used to change bound variables, using implicit substitution. Usage of this theorem is discouraged because it depends on ax-13 2386. See cbv2w 2353 with disjoint variable conditions, not depending on ax-13 2386. (Contributed by NM, 5-Aug-1993.) (Revised by Mario Carneiro, 3-Oct-2016.) Format hypotheses to common style, avoid ax-10 2141. (Revised by Wolf Lammen, 10-Sep-2023.) (New usage is discouraged.)
𝑥𝜑    &   𝑦𝜑    &   (𝜑 → Ⅎ𝑦𝜓)    &   (𝜑 → Ⅎ𝑥𝜒)    &   (𝜑 → (𝑥 = 𝑦 → (𝜓𝜒)))       (𝜑 → (∀𝑥𝜓 ↔ ∀𝑦𝜒))

Theoremcbv3h 2420 Rule used to change bound variables, using implicit substitution. Usage of this theorem is discouraged because it depends on ax-13 2386. Use the weaker cbv3hv 2356 if possible. (Contributed by NM, 8-Jun-1993.) (Proof shortened by Andrew Salmon, 25-May-2011.) (Proof shortened by Wolf Lammen, 12-May-2018.) (New usage is discouraged.)
(𝜑 → ∀𝑦𝜑)    &   (𝜓 → ∀𝑥𝜓)    &   (𝑥 = 𝑦 → (𝜑𝜓))       (∀𝑥𝜑 → ∀𝑦𝜓)

Theoremcbv1h 2421 Rule used to change bound variables, using implicit substitution. Usage of this theorem is discouraged because it depends on ax-13 2386. (Contributed by NM, 11-May-1993.) (Proof shortened by Wolf Lammen, 13-May-2018.) (New usage is discouraged.)
(𝜑 → (𝜓 → ∀𝑦𝜓))    &   (𝜑 → (𝜒 → ∀𝑥𝜒))    &   (𝜑 → (𝑥 = 𝑦 → (𝜓𝜒)))       (∀𝑥𝑦𝜑 → (∀𝑥𝜓 → ∀𝑦𝜒))

Theoremcbv2h 2422 Rule used to change bound variables, using implicit substitution. Usage of this theorem is discouraged because it depends on ax-13 2386. (Contributed by NM, 11-May-1993.) (New usage is discouraged.)
(𝜑 → (𝜓 → ∀𝑦𝜓))    &   (𝜑 → (𝜒 → ∀𝑥𝜒))    &   (𝜑 → (𝑥 = 𝑦 → (𝜓𝜒)))       (∀𝑥𝑦𝜑 → (∀𝑥𝜓 ↔ ∀𝑦𝜒))

Theoremcbv2OLD 2423 Obsolete version of cbv2 2419 as of 10-Sep-2023. (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.) (New usage is discouraged.) (Proof modification is discouraged.)
𝑥𝜑    &   𝑦𝜑    &   (𝜑 → Ⅎ𝑦𝜓)    &   (𝜑 → Ⅎ𝑥𝜒)    &   (𝜑 → (𝑥 = 𝑦 → (𝜓𝜒)))       (𝜑 → (∀𝑥𝜓 ↔ ∀𝑦𝜒))

Theoremcbvald 2424* Deduction used to change bound variables, using implicit substitution, particularly useful in conjunction with dvelim 2469. Usage of this theorem is discouraged because it depends on ax-13 2386. See cbvaldw 2354 for a version with 𝑥, 𝑦 disjoint, not depending on ax-13 2386. (Contributed by NM, 2-Jan-2002.) (Revised by Mario Carneiro, 6-Oct-2016.) (Revised by Wolf Lammen, 13-May-2018.) (New usage is discouraged.)
𝑦𝜑    &   (𝜑 → Ⅎ𝑦𝜓)    &   (𝜑 → (𝑥 = 𝑦 → (𝜓𝜒)))       (𝜑 → (∀𝑥𝜓 ↔ ∀𝑦𝜒))

Theoremcbvexd 2425* Deduction used to change bound variables, using implicit substitution, particularly useful in conjunction with dvelim 2469. Usage of this theorem is discouraged because it depends on ax-13 2386. Use the weaker cbvexdw 2355 if possible. (Contributed by NM, 2-Jan-2002.) (Revised by Mario Carneiro, 6-Oct-2016.) (New usage is discouraged.)
𝑦𝜑    &   (𝜑 → Ⅎ𝑦𝜓)    &   (𝜑 → (𝑥 = 𝑦 → (𝜓𝜒)))       (𝜑 → (∃𝑥𝜓 ↔ ∃𝑦𝜒))

Theoremcbvaldva 2426* Rule used to change the bound variable in a universal quantifier with implicit substitution. Deduction form. Usage of this theorem is discouraged because it depends on ax-13 2386. Use the weaker cbvaldvaw 2041 if possible. (Contributed by David Moews, 1-May-2017.) (New usage is discouraged.)
((𝜑𝑥 = 𝑦) → (𝜓𝜒))       (𝜑 → (∀𝑥𝜓 ↔ ∀𝑦𝜒))

Theoremcbvexdva 2427* Rule used to change the bound variable in an existential quantifier with implicit substitution. Deduction form. Usage of this theorem is discouraged because it depends on ax-13 2386. Use the weaker cbvexdvaw 2042 if possible. (Contributed by David Moews, 1-May-2017.) (New usage is discouraged.)
((𝜑𝑥 = 𝑦) → (𝜓𝜒))       (𝜑 → (∃𝑥𝜓 ↔ ∃𝑦𝜒))

Theoremcbval2 2428* Rule used to change bound variables, using implicit substitution. Usage of this theorem is discouraged because it depends on ax-13 2386. Use the weaker cbval2v 2359 if possible. (Contributed by NM, 22-Dec-2003.) (Revised by Mario Carneiro, 6-Oct-2016.) (Proof shortened by Wolf Lammen, 11-Sep-2023.) (New usage is discouraged.)
𝑧𝜑    &   𝑤𝜑    &   𝑥𝜓    &   𝑦𝜓    &   ((𝑥 = 𝑧𝑦 = 𝑤) → (𝜑𝜓))       (∀𝑥𝑦𝜑 ↔ ∀𝑧𝑤𝜓)

Theoremcbval2OLD 2429* Obsolete version of cbval2 2428 as of 11-Sep-2023. (Contributed by NM, 22-Dec-2003.) (Revised by Mario Carneiro, 6-Oct-2016.) (Proof shortened by Wolf Lammen, 22-Apr-2018.) (New usage is discouraged.) (Proof modification is discouraged.)
𝑧𝜑    &   𝑤𝜑    &   𝑥𝜓    &   𝑦𝜓    &   ((𝑥 = 𝑧𝑦 = 𝑤) → (𝜑𝜓))       (∀𝑥𝑦𝜑 ↔ ∀𝑧𝑤𝜓)

Theoremcbvex2 2430* Rule used to change bound variables, using implicit substitution. Usage of this theorem is discouraged because it depends on ax-13 2386. Use the weaker cbvex2v 2361 if possible. (Contributed by NM, 14-Sep-2003.) (Revised by Mario Carneiro, 6-Oct-2016.) (Proof shortened by Wolf Lammen, 16-Jun-2019.) (New usage is discouraged.)
𝑧𝜑    &   𝑤𝜑    &   𝑥𝜓    &   𝑦𝜓    &   ((𝑥 = 𝑧𝑦 = 𝑤) → (𝜑𝜓))       (∃𝑥𝑦𝜑 ↔ ∃𝑧𝑤𝜓)

Theoremcbval2vv 2431* Rule used to change bound variables, using implicit substitution. Usage of this theorem is discouraged because it depends on ax-13 2386. Use the weaker cbval2vw 2043 if possible. (Contributed by NM, 4-Feb-2005.) Remove dependency on ax-10 2141. (Revised by Wolf Lammen, 18-Jul-2021.) (New usage is discouraged.)
((𝑥 = 𝑧𝑦 = 𝑤) → (𝜑𝜓))       (∀𝑥𝑦𝜑 ↔ ∀𝑧𝑤𝜓)

Theoremcbvex2vv 2432* Rule used to change bound variables, using implicit substitution. Usage of this theorem is discouraged because it depends on ax-13 2386. Use the weaker cbvex2vw 2044 if possible. (Contributed by NM, 26-Jul-1995.) Remove dependency on ax-10 2141. (Revised by Wolf Lammen, 18-Jul-2021.) (New usage is discouraged.)
((𝑥 = 𝑧𝑦 = 𝑤) → (𝜑𝜓))       (∃𝑥𝑦𝜑 ↔ ∃𝑧𝑤𝜓)

Theoremcbvex4v 2433* Rule used to change bound variables, using implicit substitution. Usage of this theorem is discouraged because it depends on ax-13 2386. Use the weaker cbvex4vw 2045 if possible. (Contributed by NM, 26-Jul-1995.) (New usage is discouraged.)
((𝑥 = 𝑣𝑦 = 𝑢) → (𝜑𝜓))    &   ((𝑧 = 𝑓𝑤 = 𝑔) → (𝜓𝜒))       (∃𝑥𝑦𝑧𝑤𝜑 ↔ ∃𝑣𝑢𝑓𝑔𝜒)

Theoremequs4 2434 Lemma used in proofs of implicit substitution properties. The converse requires either a disjoint variable condition (sb56 2273) or a non-freeness hypothesis (equs45f 2478). Usage of this theorem is discouraged because it depends on ax-13 2386. See equs4v 2002 for a weaker 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.) (New usage is discouraged.)
(∀𝑥(𝑥 = 𝑦𝜑) → ∃𝑥(𝑥 = 𝑦𝜑))

Theoremequsal 2435 An equivalence related to implicit substitution. Usage of this theorem is discouraged because it depends on ax-13 2386. See equsalvw 2006 and equsalv 2263 for versions with disjoint variable conditions proved from fewer axioms. See also the dual form equsex 2436. (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.) (New usage is discouraged.)
𝑥𝜓    &   (𝑥 = 𝑦 → (𝜑𝜓))       (∀𝑥(𝑥 = 𝑦𝜑) ↔ 𝜓)

Theoremequsex 2436 An equivalence related to implicit substitution. Usage of this theorem is discouraged because it depends on ax-13 2386. See equsexvw 2007 and equsexv 2264 for versions with disjoint variable conditions proved from fewer axioms. See also the dual form equsal 2435. See equsexALT 2437 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.) (New usage is discouraged.)
𝑥𝜓    &   (𝑥 = 𝑦 → (𝜑𝜓))       (∃𝑥(𝑥 = 𝑦𝜑) ↔ 𝜓)

TheoremequsexALT 2437 Alternate proof of equsex 2436. This proves the result directly, instead of as a corollary of equsal 2435 via equs4 2434. Note in particular that only existential quantifiers appear in the proof and that the only step requiring ax-13 2386 is ax6e 2397. This proof mimics that of equsal 2435 (in particular, note that pm5.32i 577, exbii 1844, 19.41 2232, mpbiran 707 correspond respectively to pm5.74i 273, albii 1816, 19.23 2206, a1bi 365). (Contributed by BJ, 20-Aug-2020.) (Proof modification is discouraged.) (New usage is discouraged.)
𝑥𝜓    &   (𝑥 = 𝑦 → (𝜑𝜓))       (∃𝑥(𝑥 = 𝑦𝜑) ↔ 𝜓)

Theoremequsalh 2438 An equivalence related to implicit substitution. Usage of this theorem is discouraged because it depends on ax-13 2386. See equsalhw 2295 for a version with a disjoint variable condition requiring fewer axioms. (Contributed by NM, 2-Jun-1993.) (New usage is discouraged.)
(𝜓 → ∀𝑥𝜓)    &   (𝑥 = 𝑦 → (𝜑𝜓))       (∀𝑥(𝑥 = 𝑦𝜑) ↔ 𝜓)

Theoremequsexh 2439 An equivalence related to implicit substitution. Usage of this theorem is discouraged because it depends on ax-13 2386. See equsexhv 2296 for a version with a disjoint variable condition which does not require ax-13 2386. (Contributed by NM, 5-Aug-1993.) (New usage is discouraged.)
(𝜓 → ∀𝑥𝜓)    &   (𝑥 = 𝑦 → (𝜑𝜓))       (∃𝑥(𝑥 = 𝑦𝜑) ↔ 𝜓)

Theoremaxc15 2440 Derivation of set.mm's original ax-c15 36019 from ax-c11n 36018 and the shorter ax-12 2172 that has replaced it.

Theorem ax12 2441 shows the reverse derivation of ax-12 2172 from ax-c15 36019.

Normally, axc15 2440 should be used rather than ax-c15 36019, except by theorems specifically studying the latter's properties. Usage of this theorem is discouraged because it depends on ax-13 2386. (Contributed by NM, 2-Feb-2007.) (Proof shortened by Wolf Lammen, 26-Mar-2023.) (New usage is discouraged.)

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

Theoremax12 2441 Rederivation of axiom ax-12 2172 from ax12v 2173 (used only via sp 2177) , axc11r 2382, and axc15 2440 (on top of Tarski's FOL). Since this version depends on ax-13 2386, usage of the weaker ax12v 2173, ax12w 2133, ax12i 1965 are preferred. (Contributed by NM, 22-Jan-2007.) Proof uses contemporary axioms. (Revised by Wolf Lammen, 8-Aug-2020.) (Proof shortened by BJ, 4-Jul-2021.) (New usage is discouraged.)
(𝑥 = 𝑦 → (∀𝑦𝜑 → ∀𝑥(𝑥 = 𝑦𝜑)))

Theoremax12b 2442 A bidirectional version of axc15 2440. Usage of this theorem is discouraged because it depends on ax-13 2386. (Contributed by NM, 30-Jun-2006.) (New usage is discouraged.)
((¬ ∀𝑥 𝑥 = 𝑦𝑥 = 𝑦) → (𝜑 ↔ ∀𝑥(𝑥 = 𝑦𝜑)))

Theoremax13ALT 2443 Alternate proof of ax13 2389 from FOL, sp 2177, and axc9 2396. (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 2444 Derive set.mm's original ax-c11n 36018 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 2056. Use aecom 2445 instead when this does not lengthen the proof. Usage of this theorem is discouraged because it depends on ax-13 2386. (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.) (New usage is discouraged.)
(∀𝑥 𝑥 = 𝑦 → ∀𝑦 𝑦 = 𝑥)

Theoremaecom 2445 Commutation law for identical variable specifiers. Both sides of the biconditional are true when 𝑥 and 𝑦 are substituted with the same variable. Usage of this theorem is discouraged because it depends on ax-13 2386. (Contributed by NM, 10-May-1993.) Change to a biconditional. (Revised by BJ, 26-Sep-2019.) (New usage is discouraged.)
(∀𝑥 𝑥 = 𝑦 ↔ ∀𝑦 𝑦 = 𝑥)

Theoremaecoms 2446 A commutation rule for identical variable specifiers. Usage of this theorem is discouraged because it depends on ax-13 2386. (Contributed by NM, 10-May-1993.) (New usage is discouraged.)
(∀𝑥 𝑥 = 𝑦𝜑)       (∀𝑦 𝑦 = 𝑥𝜑)

Theoremnaecoms 2447 A commutation rule for distinct variable specifiers. Usage of this theorem is discouraged because it depends on ax-13 2386. (Contributed by NM, 2-Jan-2002.) (New usage is discouraged.)
(¬ ∀𝑥 𝑥 = 𝑦𝜑)       (¬ ∀𝑦 𝑦 = 𝑥𝜑)

Theoremaxc11 2448 Show that ax-c11 36017 can be derived from ax-c11n 36018 in the form of axc11n 2444. Normally, axc11 2448 should be used rather than ax-c11 36017, except by theorems specifically studying the latter's properties. Usage of this theorem is discouraged because it depends on ax-13 2386. Use the weaker axc11v 2260 when possible. (Contributed by NM, 16-May-2008.) (Proof shortened by Wolf Lammen, 21-Apr-2018.) (New usage is discouraged.)
(∀𝑥 𝑥 = 𝑦 → (∀𝑥𝜑 → ∀𝑦𝜑))

Theoremhbae 2449 All variables are effectively bound in an identical variable specifier. Usage of this theorem is discouraged because it depends on ax-13 2386. Use the weaker hbaev 2060 when possible. (Contributed by NM, 13-May-1993.) (Proof shortened by Wolf Lammen, 21-Apr-2018.) (New usage is discouraged.)
(∀𝑥 𝑥 = 𝑦 → ∀𝑧𝑥 𝑥 = 𝑦)

Theoremhbnae 2450 All variables are effectively bound in a distinct variable specifier. Lemma L19 in [Megill] p. 446 (p. 14 of the preprint). Usage of this theorem is discouraged because it depends on ax-13 2386. Use the weaker hbnaev 2063 when possible. (Contributed by NM, 13-May-1993.) (New usage is discouraged.)
(¬ ∀𝑥 𝑥 = 𝑦 → ∀𝑧 ¬ ∀𝑥 𝑥 = 𝑦)

Theoremnfae 2451 All variables are effectively bound in an identical variable specifier. Usage of this theorem is discouraged because it depends on ax-13 2386. (Contributed by Mario Carneiro, 11-Aug-2016.) (New usage is discouraged.)
𝑧𝑥 𝑥 = 𝑦

Theoremnfnae 2452 All variables are effectively bound in a distinct variable specifier. See also nfnaew 2149. Usage of this theorem is discouraged because it depends on ax-13 2386. Use the weaker nfnaew 2149 when possible. (Contributed by Mario Carneiro, 11-Aug-2016.) (New usage is discouraged.)
𝑧 ¬ ∀𝑥 𝑥 = 𝑦

Theoremhbnaes 2453 Rule that applies hbnae 2450 to antecedent. Usage of this theorem is discouraged because it depends on ax-13 2386. (Contributed by NM, 15-May-1993.) (New usage is discouraged.)
(∀𝑧 ¬ ∀𝑥 𝑥 = 𝑦𝜑)       (¬ ∀𝑥 𝑥 = 𝑦𝜑)

Theoremaxc16i 2454* Inference with axc16 2257 as its conclusion. Usage of axc16 2257 is preferred since it requires fewer axioms. (Contributed by NM, 20-May-2008.) (Proof modification is discouraged.) (New usage is discouraged.)
(𝑥 = 𝑧 → (𝜑𝜓))    &   (𝜓 → ∀𝑥𝜓)       (∀𝑥 𝑥 = 𝑦 → (𝜑 → ∀𝑥𝜑))

Theoremaxc16nfALT 2455* Alternate proof of axc16nf 2259, shorter but requiring ax-11 2156 and ax-13 2386. (Contributed by Mario Carneiro, 7-Oct-2016.) (Proof modification is discouraged.) (New usage is discouraged.)
(∀𝑥 𝑥 = 𝑦 → Ⅎ𝑧𝜑)

Theoremdral2 2456 Formula-building lemma for use with the Distinctor Reduction Theorem. Part of Theorem 9.4 of [Megill] p. 448 (p. 16 of preprint). Usage of this theorem is discouraged because it depends on ax-13 2386. Usage of albidv 1917 is preferred, which requires fewer axioms. (Contributed by NM, 27-Feb-2005.) Allow a shortening of dral1 2457. (Revised by Wolf Lammen, 4-Mar-2018.) (New usage is discouraged.)
(∀𝑥 𝑥 = 𝑦 → (𝜑𝜓))       (∀𝑥 𝑥 = 𝑦 → (∀𝑧𝜑 ↔ ∀𝑧𝜓))

Theoremdral1 2457 Formula-building lemma for use with the Distinctor Reduction Theorem. Part of Theorem 9.4 of [Megill] p. 448 (p. 16 of preprint). Usage of this theorem is discouraged because it depends on ax-13 2386. Use the weaker dral1v 2383 if possible. (Contributed by NM, 24-Nov-1994.) Remove dependency on ax-11 2156. (Revised by Wolf Lammen, 6-Sep-2018.) (New usage is discouraged.)
(∀𝑥 𝑥 = 𝑦 → (𝜑𝜓))       (∀𝑥 𝑥 = 𝑦 → (∀𝑥𝜑 ↔ ∀𝑦𝜓))

Theoremdral1ALT 2458 Alternate proof of dral1 2457, shorter but requiring ax-11 2156. (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 2459 Formula-building lemma for use with the Distinctor Reduction Theorem. Part of Theorem 9.4 of [Megill] p. 448 (p. 16 of preprint). Usage of this theorem is discouraged because it depends on ax-13 2386. Use the weaker drex1v 2384 if possible. (Contributed by NM, 27-Feb-2005.) (New usage is discouraged.)
(∀𝑥 𝑥 = 𝑦 → (𝜑𝜓))       (∀𝑥 𝑥 = 𝑦 → (∃𝑥𝜑 ↔ ∃𝑦𝜓))

Theoremdrex2 2460 Formula-building lemma for use with the Distinctor Reduction Theorem. Part of Theorem 9.4 of [Megill] p. 448 (p. 16 of preprint). Usage of this theorem is discouraged because it depends on ax-13 2386. Usage of exbidv 1918 is preferred, which requires fewer axioms. (Contributed by NM, 27-Feb-2005.) (New usage is discouraged.)
(∀𝑥 𝑥 = 𝑦 → (𝜑𝜓))       (∀𝑥 𝑥 = 𝑦 → (∃𝑧𝜑 ↔ ∃𝑧𝜓))

Theoremdrnf1 2461 Formula-building lemma for use with the Distinctor Reduction Theorem. Usage of this theorem is discouraged because it depends on ax-13 2386. Use the weaker drnf1v 2385 if possible. (Contributed by Mario Carneiro, 4-Oct-2016.) (New usage is discouraged.)
(∀𝑥 𝑥 = 𝑦 → (𝜑𝜓))       (∀𝑥 𝑥 = 𝑦 → (Ⅎ𝑥𝜑 ↔ Ⅎ𝑦𝜓))

Theoremdrnf2 2462 Formula-building lemma for use with the Distinctor Reduction Theorem. Usage of this theorem is discouraged because it depends on ax-13 2386. Usage of nfbidv 1919 is preferred, which requires fewer axioms. (Contributed by Mario Carneiro, 4-Oct-2016.) (Proof shortened by Wolf Lammen, 5-May-2018.) (New usage is discouraged.)
(∀𝑥 𝑥 = 𝑦 → (𝜑𝜓))       (∀𝑥 𝑥 = 𝑦 → (Ⅎ𝑧𝜑 ↔ Ⅎ𝑧𝜓))

Theoremnfald2 2463 Variation on nfald 2343 which adds the hypothesis that 𝑥 and 𝑦 are distinct in the inner subproof. Usage of this theorem is discouraged because it depends on ax-13 2386. Check out nfald 2343 for a version requiring fewer axioms. (Contributed by Mario Carneiro, 8-Oct-2016.) (New usage is discouraged.)
𝑦𝜑    &   ((𝜑 ∧ ¬ ∀𝑥 𝑥 = 𝑦) → Ⅎ𝑥𝜓)       (𝜑 → Ⅎ𝑥𝑦𝜓)

Theoremnfexd2 2464 Variation on nfexd 2344 which adds the hypothesis that 𝑥 and 𝑦 are distinct in the inner subproof. Usage of this theorem is discouraged because it depends on ax-13 2386. Check out nfexd 2344 for a version requiring fewer axioms. (Contributed by Mario Carneiro, 8-Oct-2016.) (New usage is discouraged.)
𝑦𝜑    &   ((𝜑 ∧ ¬ ∀𝑥 𝑥 = 𝑦) → Ⅎ𝑥𝜓)       (𝜑 → Ⅎ𝑥𝑦𝜓)

Theoremexdistrf 2465 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). Usage of this theorem is discouraged because it depends on ax-13 2386. Check out exdistr 1951 for a version requiring fewer axioms. (Contributed by Mario Carneiro, 20-Mar-2013.) (Proof shortened by Wolf Lammen, 14-May-2018.) (New usage is discouraged.)
(¬ ∀𝑥 𝑥 = 𝑦 → Ⅎ𝑦𝜑)       (∃𝑥𝑦(𝜑𝜓) → ∃𝑥(𝜑 ∧ ∃𝑦𝜓))

Theoremdvelimf 2466 Version of dvelimv 2470 without any variable restrictions. Usage of this theorem is discouraged because it depends on ax-13 2386. (Contributed by NM, 1-Oct-2002.) (Revised by Mario Carneiro, 6-Oct-2016.) (Proof shortened by Wolf Lammen, 11-May-2018.) (New usage is discouraged.)
𝑥𝜑    &   𝑧𝜓    &   (𝑧 = 𝑦 → (𝜑𝜓))       (¬ ∀𝑥 𝑥 = 𝑦 → Ⅎ𝑥𝜓)

Theoremdvelimdf 2467 Deduction form of dvelimf 2466. Usage of this theorem is discouraged because it depends on ax-13 2386. (Contributed by NM, 7-Apr-2004.) (Revised by Mario Carneiro, 6-Oct-2016.) (Proof shortened by Wolf Lammen, 11-May-2018.) (New usage is discouraged.)
𝑥𝜑    &   𝑧𝜑    &   (𝜑 → Ⅎ𝑥𝜓)    &   (𝜑 → Ⅎ𝑧𝜒)    &   (𝜑 → (𝑧 = 𝑦 → (𝜓𝜒)))       (𝜑 → (¬ ∀𝑥 𝑥 = 𝑦 → Ⅎ𝑥𝜒))

Theoremdvelimh 2468 Version of dvelim 2469 without any variable restrictions. Usage of this theorem is discouraged because it depends on ax-13 2386. Check out dvelimhw 2362 for a version requiring fewer axioms. (Contributed by NM, 1-Oct-2002.) (Proof shortened by Wolf Lammen, 11-May-2018.) (New usage is discouraged.)
(𝜑 → ∀𝑥𝜑)    &   (𝜓 → ∀𝑧𝜓)    &   (𝑧 = 𝑦 → (𝜑𝜓))       (¬ ∀𝑥 𝑥 = 𝑦 → (𝜓 → ∀𝑥𝜓))

Theoremdvelim 2469* 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 2467.

Other variants of this theorem are dvelimh 2468 (with no distinct variable restrictions) and dvelimhw 2362 (that avoids ax-13 2386). Usage of this theorem is discouraged because it depends on ax-13 2386. Check out dvelimhw 2362 for a version requiring fewer axioms. (Contributed by NM, 23-Nov-1994.) (New usage is discouraged.)

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

Theoremdvelimv 2470* Similar to dvelim 2469 with first hypothesis replaced by a distinct variable condition. Usage of this theorem is discouraged because it depends on ax-13 2386. Check out dvelimhw 2362 for a version requiring fewer axioms. (Contributed by NM, 25-Jul-2015.) (Proof shortened by Wolf Lammen, 30-Apr-2018.) (New usage is discouraged.)
(𝑧 = 𝑦 → (𝜑𝜓))       (¬ ∀𝑥 𝑥 = 𝑦 → (𝜓 → ∀𝑥𝜓))

Theoremdvelimnf 2471* Version of dvelim 2469 using "not free" notation. Usage of this theorem is discouraged because it depends on ax-13 2386. (Contributed by Mario Carneiro, 9-Oct-2016.) (New usage is discouraged.)
𝑥𝜑    &   (𝑧 = 𝑦 → (𝜑𝜓))       (¬ ∀𝑥 𝑥 = 𝑦 → Ⅎ𝑥𝜓)

Theoremdveeq2ALT 2472* Alternate proof of dveeq2 2392, shorter but requiring ax-11 2156. (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))

Theoremequvini 2473 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. Usage of this theorem is discouraged because it depends on ax-13 2386. See equvinv 2032 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, 16-Sep-2023.) (New usage is discouraged.)
(𝑥 = 𝑦 → ∃𝑧(𝑥 = 𝑧𝑧 = 𝑦))

TheoremequviniOLD 2474 Obsolete version of equvini 2473 as of 16-Sep-2023. (Contributed by NM, 10-Jan-1993.) (Proof shortened by Andrew Salmon, 25-May-2011.) (Proof shortened by Wolf Lammen, 15-Sep-2018.) (New usage is discouraged.) (Proof modification is discouraged.)
(𝑥 = 𝑦 → ∃𝑧(𝑥 = 𝑧𝑧 = 𝑦))

Theoremequvel 2475 A variable elimination law for equality with no distinct variable requirements. Compare equvini 2473. Usage of this theorem is discouraged because it depends on ax-13 2386. Use the weaker equvelv 2034 when possible. (Contributed by NM, 1-Mar-2013.) (Proof shortened by Mario Carneiro, 17-Oct-2016.) (Proof shortened by Wolf Lammen, 15-Jun-2019.) (New usage is discouraged.)
(∀𝑧(𝑧 = 𝑥𝑧 = 𝑦) → 𝑥 = 𝑦)

Theoremequs5a 2476 A property related to substitution that unlike equs5 2479 does not require a distinctor antecedent. Usage of this theorem is discouraged because it depends on ax-13 2386. See equs5av 2275 and equs5aALT 2380 for proofs using ax-12 2172 but not ax13 2389. (Contributed by NM, 2-Feb-2007.) (New usage is discouraged.)
(∃𝑥(𝑥 = 𝑦 ∧ ∀𝑦𝜑) → ∀𝑥(𝑥 = 𝑦𝜑))

Theoremequs5e 2477 A property related to substitution that unlike equs5 2479 does not require a distinctor antecedent. See equs5eALT 2381 for an alternate proof using ax-12 2172 but not ax13 2389. Usage of this theorem is discouraged because it depends on ax-13 2386. (Contributed by NM, 2-Feb-2007.) (Proof shortened by Wolf Lammen, 15-Jan-2018.) (New usage is discouraged.)
(∃𝑥(𝑥 = 𝑦𝜑) → ∀𝑥(𝑥 = 𝑦 → ∃𝑦𝜑))

Theoremequs45f 2478 Two ways of expressing substitution when 𝑦 is not free in 𝜑. The implication "to the left" is equs4 2434 and does not require the non-freeness hypothesis. Theorem sb56 2273 replaces the non-freeness hypothesis with a disjoint variable condition and equs5 2479 replaces it with a distinctor as antecedent. Usage of this theorem is discouraged because it depends on ax-13 2386. (Contributed by NM, 25-Apr-2008.) (Revised by Mario Carneiro, 4-Oct-2016.) (New usage is discouraged.)
𝑦𝜑       (∃𝑥(𝑥 = 𝑦𝜑) ↔ ∀𝑥(𝑥 = 𝑦𝜑))

Theoremequs5 2479 Lemma used in proofs of substitution properties. If there is a disjoint variable condition on 𝑥, 𝑦, then sb56 2273 can be used instead; if 𝑦 is not free in 𝜑, then equs45f 2478 can be used. Usage of this theorem is discouraged because it depends on ax-13 2386. (Contributed by NM, 14-May-1993.) (Revised by BJ, 1-Oct-2018.) (New usage is discouraged.)
(¬ ∀𝑥 𝑥 = 𝑦 → (∃𝑥(𝑥 = 𝑦𝜑) ↔ ∀𝑥(𝑥 = 𝑦𝜑)))

Theoremdveel1 2480* Quantifier introduction when one pair of variables is distinct. Usage of this theorem is discouraged because it depends on ax-13 2386. (Contributed by NM, 2-Jan-2002.) (New usage is discouraged.)
(¬ ∀𝑥 𝑥 = 𝑦 → (𝑦𝑧 → ∀𝑥 𝑦𝑧))

Theoremdveel2 2481* Quantifier introduction when one pair of variables is distinct. Usage of this theorem is discouraged because it depends on ax-13 2386. (Contributed by NM, 2-Jan-2002.) (New usage is discouraged.)
(¬ ∀𝑥 𝑥 = 𝑦 → (𝑧𝑦 → ∀𝑥 𝑧𝑦))

Theoremaxc14 2482 Axiom ax-c14 36021 is redundant if we assume ax-5 1907. Remark 9.6 in [Megill] p. 448 (p. 16 of the preprint), regarding axiom scheme C14'.

Note that 𝑤 is a dummy variable introduced in the proof. Its purpose is to satisfy the distinct variable requirements of dveel2 2481 and ax-5 1907. By the end of the proof it has vanished, and the final theorem has no distinct variable requirements. Usage of this theorem is discouraged because it depends on ax-13 2386. (Contributed by NM, 29-Jun-1995.) (Proof modification is discouraged.) (New usage is discouraged.)

(¬ ∀𝑧 𝑧 = 𝑥 → (¬ ∀𝑧 𝑧 = 𝑦 → (𝑥𝑦 → ∀𝑧 𝑥𝑦)))

Theoremsb6x 2483 Equivalence involving substitution for a variable not free. Usage of this theorem is discouraged because it depends on ax-13 2386. Usage of sb6 2089 is preferred, which requires fewer axioms. (Contributed by NM, 2-Jun-1993.) (Revised by Mario Carneiro, 4-Oct-2016.) (New usage is discouraged.)
𝑥𝜑       ([𝑦 / 𝑥]𝜑 ↔ ∀𝑥(𝑥 = 𝑦𝜑))

Theoremsbequ5 2484 Substitution does not change an identical variable specifier. Usage of this theorem is discouraged because it depends on ax-13 2386. (Contributed by NM, 15-May-1993.) (New usage is discouraged.)
([𝑤 / 𝑧]∀𝑥 𝑥 = 𝑦 ↔ ∀𝑥 𝑥 = 𝑦)

Theoremsbequ6 2485 Substitution does not change a distinctor. Usage of this theorem is discouraged because it depends on ax-13 2386. (Contributed by NM, 5-Aug-1993.) (New usage is discouraged.)
([𝑤 / 𝑧] ¬ ∀𝑥 𝑥 = 𝑦 ↔ ¬ ∀𝑥 𝑥 = 𝑦)

Theoremsb5rf 2486 Reversed substitution. Usage of this theorem is discouraged because it depends on ax-13 2386. (Contributed by NM, 3-Feb-2005.) (Revised by Mario Carneiro, 6-Oct-2016.) (Proof shortened by Wolf Lammen, 20-Sep-2018.) (New usage is discouraged.)
𝑦𝜑       (𝜑 ↔ ∃𝑦(𝑦 = 𝑥 ∧ [𝑦 / 𝑥]𝜑))

Theoremsb6rf 2487 Reversed substitution. For a version requiring disjoint variables, but fewer axioms, see sb6rfv 2372. Usage of this theorem is discouraged because it depends on ax-13 2386. Use the weaker sb6rfv 2372 if possible. (Contributed by NM, 1-Aug-1993.) (Revised by Mario Carneiro, 6-Oct-2016.) (Proof shortened by Wolf Lammen, 21-Sep-2018.) (New usage is discouraged.)
𝑦𝜑       (𝜑 ↔ ∀𝑦(𝑦 = 𝑥 → [𝑦 / 𝑥]𝜑))

Theoremax12vALT 2488* Alternate proof of ax12v2 2174, shorter, but depending on more axioms. (Contributed by NM, 5-Aug-1993.) (New usage is discouraged.) (Proof modification is discouraged.)
(𝑥 = 𝑦 → (𝜑 → ∀𝑥(𝑥 = 𝑦𝜑)))

Theorem2ax6elem 2489 We can always find values matching 𝑥 and 𝑦, as long as they are represented by distinct variables. This theorem merges two ax6e 2397 instances 𝑧𝑧 = 𝑥 and 𝑤𝑤 = 𝑦 into a common expression. Alan Sare contributed a variant of this theorem with distinct variable conditions before, see ax6e2nd 40885. Usage of this theorem is discouraged because it depends on ax-13 2386. (Contributed by Wolf Lammen, 27-Sep-2018.) (New usage is discouraged.)
(¬ ∀𝑤 𝑤 = 𝑧 → ∃𝑧𝑤(𝑧 = 𝑥𝑤 = 𝑦))

Theorem2ax6e 2490* We can always find values matching 𝑥 and 𝑦, as long as they are represented by distinct variables. Version of 2ax6elem 2489 with a distinct variable constraint. Usage of this theorem is discouraged because it depends on ax-13 2386. (Contributed by Wolf Lammen, 28-Sep-2018.) (Proof shortened by Wolf Lammen, 3-Oct-2023.) (New usage is discouraged.)
𝑧𝑤(𝑧 = 𝑥𝑤 = 𝑦)

Theorem2ax6eOLD 2491* Obsolete version of 2ax6e 2490 as of 3-Oct-2023. (Contributed by Wolf Lammen, 28-Sep-2018.) (Proof modification is discouraged.) (New usage is discouraged.)
𝑧𝑤(𝑧 = 𝑥𝑤 = 𝑦)

Theorem2sb5rf 2492* Reversed double substitution. Usage of this theorem is discouraged because it depends on ax-13 2386. (Contributed by NM, 3-Feb-2005.) (Revised by Mario Carneiro, 6-Oct-2016.) Remove distinct variable constraints. (Revised by Wolf Lammen, 28-Sep-2018.) (New usage is discouraged.)
𝑧𝜑    &   𝑤𝜑       (𝜑 ↔ ∃𝑧𝑤((𝑧 = 𝑥𝑤 = 𝑦) ∧ [𝑧 / 𝑥][𝑤 / 𝑦]𝜑))

Theorem2sb6rf 2493* Reversed double substitution. Usage of this theorem is discouraged because it depends on ax-13 2386. (Contributed by NM, 3-Feb-2005.) (Revised by Mario Carneiro, 6-Oct-2016.) Remove variable constraints. (Revised by Wolf Lammen, 28-Sep-2018.) (Proof shortened by Wolf Lammen, 13-Apr-2023.) (New usage is discouraged.)
𝑧𝜑    &   𝑤𝜑       (𝜑 ↔ ∀𝑧𝑤((𝑧 = 𝑥𝑤 = 𝑦) → [𝑧 / 𝑥][𝑤 / 𝑦]𝜑))

Theoremsbel2x 2494* Elimination of double substitution. Usage of this theorem is discouraged because it depends on ax-13 2386. (Contributed by NM, 5-Aug-1993.) (Proof shortened by Wolf Lammen, 29-Sep-2018.) (New usage is discouraged.)
(𝜑 ↔ ∃𝑥𝑦((𝑥 = 𝑧𝑦 = 𝑤) ∧ [𝑦 / 𝑤][𝑥 / 𝑧]𝜑))

Theoremsb4b 2495 Simplified definition of substitution when variables are distinct. Version of sb6 2089 with a distinctor. Usage of this theorem is discouraged because it depends on ax-13 2386. (Contributed by NM, 27-May-1997.) Revise df-sb 2066. (Revised by Wolf Lammen, 21-Feb-2024.) (New usage is discouraged.)
(¬ ∀𝑥 𝑥 = 𝑡 → ([𝑡 / 𝑥]𝜑 ↔ ∀𝑥(𝑥 = 𝑡𝜑)))

Theoremsb4bOLD 2496 Obsolete version of sb4b 2495 as of 21-Feb-2024. (Contributed by NM, 27-May-1997.) Revise df-sb 2066. (Revised by Wolf Lammen, 25-Jul-2023.) (New usage is discouraged.) (Proof modification is discouraged.)
(¬ ∀𝑥 𝑥 = 𝑡 → ([𝑡 / 𝑥]𝜑 ↔ ∀𝑥(𝑥 = 𝑡𝜑)))

Theoremsb3b 2497 Simplified definition of substitution when variables are distinct. This is the biconditional strengthening of sb3 2498. Usage of this theorem is discouraged because it depends on ax-13 2386. (Contributed by BJ, 6-Oct-2018.) Shorten sb3 2498. (Revised by Wolf Lammen, 21-Feb-2021.) (New usage is discouraged.)
(¬ ∀𝑥 𝑥 = 𝑦 → ([𝑦 / 𝑥]𝜑 ↔ ∃𝑥(𝑥 = 𝑦𝜑)))

Theoremsb3 2498 One direction of a simplified definition of substitution when variables are distinct. Usage of this theorem is discouraged because it depends on ax-13 2386. (Contributed by NM, 5-Aug-1993.) (Proof shortened by Wolf Lammen, 21-Feb-2024.) (New usage is discouraged.)
(¬ ∀𝑥 𝑥 = 𝑦 → (∃𝑥(𝑥 = 𝑦𝜑) → [𝑦 / 𝑥]𝜑))

Theoremsb1 2499 One direction of a simplified definition of substitution. The converse requires either a disjoint variable condition (sb5 2272) or a non-freeness hypothesis (sb5f 2534). Usage of this theorem is discouraged because it depends on ax-13 2386. Use the weaker sb1v 2091 when possible. (Contributed by NM, 13-May-1993.) Revise df-sb 2066. (Revised by Wolf Lammen, 21-Feb-2024.) (New usage is discouraged.)
([𝑦 / 𝑥]𝜑 → ∃𝑥(𝑥 = 𝑦𝜑))

Theoremsb2 2500 One direction of a simplified definition of substitution. The converse requires either a disjoint variable condition (sb6 2089) or a non-freeness hypothesis (sb6f 2533). Usage of this theorem is discouraged because it depends on ax-13 2386. (Contributed by NM, 13-May-1993.) Revise df-sb 2066. (Revised by Wolf Lammen, 26-Jul-2023.) (New usage is discouraged.)
(∀𝑥(𝑥 = 𝑦𝜑) → [𝑦 / 𝑥]𝜑)

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