P. 25 of Theorem List - Metamath Proof Explorer

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

Type	Label	Description
Statement
Theorem	equs5aALT 2401	Alternate proof of equs5a 2492. Uses ax-12 2216 but not ax-13 2407. (Contributed by NM, 2-Feb-2007.) (Proof modification is discouraged.) (New usage is discouraged.)
⊢ (∃𝑥(𝑥 = 𝑦 ∧ ∀𝑦𝜑) → ∀𝑥(𝑥 = 𝑦 → 𝜑))
Theorem	equs5eALT 2402	Alternate proof of equs5e 2493. Uses ax-12 2216 but not ax-13 2407. (Contributed by NM, 2-Feb-2007.) (Proof shortened by Wolf Lammen, 15-Jan-2018.) (Proof modification is discouraged.) (New usage is discouraged.)
⊢ (∃𝑥(𝑥 = 𝑦 ∧ 𝜑) → ∀𝑥(𝑥 = 𝑦 → ∃𝑦𝜑))
Theorem	axc11r 2403	Same as axc11 2465 but with reversed antecedent. Note the use of ax-12 2216 (and not merely ax12v 2217 as in axc11rv 2304). This theorem is mostly used to eliminate conditions requiring set variables be distinct (cf. cbvaev 2088 and aecom 2462, for example) in proofs. In practice, theorems beyond elementary set theory do not really benefit from such eliminations. As of 2024, it is used in conjunction with ax-13 2407 only, and like that, it should be applied only in niches where indispensable. (Contributed by NM, 25-Jul-2015.)
⊢ (∀𝑦 𝑦 = 𝑥 → (∀𝑥𝜑 → ∀𝑦𝜑))
Theorem	dral1v 2404*	Formula-building lemma for use with the Distinctor Reduction Theorem. Version of dral1 2474 with a disjoint variable condition, which does not require ax-13 2407. Remark: the corresponding versions for dral2 2473 and drex2 2477 are instances of albidv 1953 and exbidv 1954 respectively. (Contributed by NM, 24-Nov-1994.) (Revised by BJ, 17-Jun-2019.) Base the proof on ax12v 2217. (Revised by Wolf Lammen, 30-Mar-2024.) Avoid ax-10 2179. (Revised by GG, 18-Nov-2024.)
⊢ (∀𝑥 𝑥 = 𝑦 → (𝜑 ↔ 𝜓))    ⇒   ⊢ (∀𝑥 𝑥 = 𝑦 → (∀𝑥𝜑 ↔ ∀𝑦𝜓))
Theorem	drex1v 2405*	Formula-building lemma for use with the Distinctor Reduction Theorem. Version of drex1 2476 with a disjoint variable condition, which does not require ax-13 2407. (Contributed by NM, 27-Feb-2005.) (Revised by BJ, 17-Jun-2019.)
⊢ (∀𝑥 𝑥 = 𝑦 → (𝜑 ↔ 𝜓))    ⇒   ⊢ (∀𝑥 𝑥 = 𝑦 → (∃𝑥𝜑 ↔ ∃𝑦𝜓))
Theorem	drnf1v 2406*	Formula-building lemma for use with the Distinctor Reduction Theorem. Version of drnf1 2478 with a disjoint variable condition, which does not require ax-13 2407. (Contributed by Mario Carneiro, 4-Oct-2016.) (Revised by BJ, 17-Jun-2019.) Avoid ax-10 2179. (Revised by GG, 18-Nov-2024.)
⊢ (∀𝑥 𝑥 = 𝑦 → (𝜑 ↔ 𝜓))    ⇒   ⊢ (∀𝑥 𝑥 = 𝑦 → (Ⅎ𝑥𝜑 ↔ Ⅎ𝑦𝜓))
1.5.4  Axiom scheme ax-13 (Quantified Equality)
Axiom	ax-13 2407	Axiom of Quantified Equality. One of the equality and substitution axioms of predicate calculus with equality. An equivalent way to express this axiom that may be easier to understand is (¬ 𝑥 = 𝑦 → (¬ 𝑥 = 𝑧 → (𝑦 = 𝑧 → ∀𝑥𝑦 = 𝑧))) (see ax13b 2065). Recall that in the intended interpretation, our variables are metavariables ranging over the variables of predicate calculus (the object language). In order for the first antecedent ¬ 𝑥 = 𝑦 to hold, 𝑥 and 𝑦 must have different values and thus cannot be the same object-language variable (so they are effectively "distinct variables" even though no $d is present). Similarly, 𝑥 and 𝑧 cannot be the same object-language variable. Therefore, 𝑥 will not occur in the wff 𝑦 = 𝑧 when the first two antecedents hold, so analogous to ax-5 1943, the conclusion (𝑦 = 𝑧 → ∀𝑥𝑦 = 𝑧) follows. Note that ax-5 1943 cannot prove this because its distinct variable ($d) requirement is not satisfied directly but only indirectly (outside of Metamath) by the argument above. The original version of this axiom was ax-c9 39704 and was replaced with this shorter ax-13 2407 in December 2015. The old axiom is proved from this one as Theorem axc9 2417. The primary purpose of this axiom is to provide a way to introduce the quantifier ∀𝑥 on 𝑦 = 𝑧 even when 𝑥 and 𝑦 are substituted with the same variable. In this case, the first antecedent becomes ¬ 𝑥 = 𝑥 and the axiom still holds. This axiom is mostly used to eliminate conditions requiring set variables be distinct (cf. ax6ev 2002 and ax6e 2418, for example) in proofs. In practice, theorems beyond elementary set theory do not really benefit from such eliminations, so direct or indirect application of this axiom is discouraged now. You need to explicitly confirm its use in case you see a sensible application in a niche. After some assisting contributions by others over the years, it was in particular the extensive work of Gino Giotto in 2024 that helped reducing dependencies on this axiom on a large scale. Although this version is shorter, the original version axc9 2417 may be more practical to work with because of the "distinctor" form of its antecedents. A typical application of axc9 2417 is in dvelimh 2485 which converts a distinct variable pair to the distinctor antecedent ¬ ∀𝑥𝑥 = 𝑦. In particular, it is conjectured that it is not possible to prove ax6 2419 from ax6v 2001 without this axiom. This axiom can be weakened if desired by adding distinct variable restrictions on pairs 𝑥, 𝑧 and 𝑦, 𝑧. To show that, we add these restrictions to Theorem ax13v 2408 and use only ax13v 2408 for further derivations. Thus, ax13v 2408 should be the only theorem referencing this axiom. Other theorems can reference either ax13v 2408 (preferred) or ax13 2410 (if the stronger form is needed). This axiom scheme is logically redundant (see ax13w 2174) but is used as an auxiliary axiom scheme to achieve scheme completeness (i.e. so that all possible cases of bundling can be proved; see text linked at mmtheorems.html#ax6dgen 2174). It is not known whether this axiom can be derived from the others. (Contributed by NM, 21-Dec-2015.) (New usage is discouraged.)
⊢ (¬ 𝑥 = 𝑦 → (𝑦 = 𝑧 → ∀𝑥 𝑦 = 𝑧))
Theorem	ax13v 2408*	A weaker version of ax-13 2407 with distinct variable restrictions on pairs 𝑥, 𝑧 and 𝑦, 𝑧. In order to show (with ax13 2410) that this weakening is still adequate, this should be the only theorem referencing ax-13 2407 directly. Had we additionally required 𝑥 and 𝑦 be distinct, too, this theorem would have been a direct consequence of ax-5 1943. So essentially this theorem states, that a distinct variable condition can be replaced with an inequality between set variables. Preferably, use the version ax13w 2174 to avoid the propagation of ax-13 2407. (Contributed by NM, 30-Jun-2016.) (New usage is discouraged.)
⊢ (¬ 𝑥 = 𝑦 → (𝑦 = 𝑧 → ∀𝑥 𝑦 = 𝑧))
Theorem	ax13lem1 2409*	A version of ax13v 2408 with one distinct variable restriction dropped. For convenience, 𝑦 is kept on the right side of equations. The proof of ax13 2410 bases on ideas from NM, 24-Dec-2015. (Contributed by Wolf Lammen, 8-Sep-2018.) (New usage is discouraged.)
⊢ (¬ 𝑥 = 𝑦 → (𝑧 = 𝑦 → ∀𝑥 𝑧 = 𝑦))
Theorem	ax13 2410	Derive ax-13 2407 from ax13v 2408 and Tarski's FOL. This shows that the weakening in ax13v 2408 is still sufficient for a complete system. Preferably, use the weaker ax13w 2174 to avoid the propagation of ax-13 2407. (Contributed by NM, 21-Dec-2015.) (Proof shortened by Wolf Lammen, 31-Jan-2018.) Reduce axiom usage. (Revised by Wolf Lammen, 2-Jun-2021.) (New usage is discouraged.)
⊢ (¬ 𝑥 = 𝑦 → (𝑦 = 𝑧 → ∀𝑥 𝑦 = 𝑧))
Theorem	ax13lem2 2411*	Lemma for nfeqf2 2412. This lemma is equivalent to ax13v 2408 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 = 𝑦 → 𝑧 = 𝑦))
Theorem	nfeqf2 2412*	An equation between setvar is free of any other setvar. Usage of this theorem is discouraged because it depends on ax-13 2407. (Contributed by Wolf Lammen, 9-Jun-2019.) Remove dependency on ax-12 2216. (Revised by Wolf Lammen, 16-Dec-2022.) (New usage is discouraged.)
⊢ (¬ ∀𝑥 𝑥 = 𝑦 → Ⅎ𝑥 𝑧 = 𝑦)
Theorem	dveeq2 2413*	Quantifier introduction when one pair of variables is distinct. Usage of this theorem is discouraged because it depends on ax-13 2407. (Contributed by NM, 2-Jan-2002.) (Revised by NM, 20-Jul-2015.) Remove dependency on ax-11 2195. (Revised by Wolf Lammen, 8-Sep-2018.) (New usage is discouraged.)
⊢ (¬ ∀𝑥 𝑥 = 𝑦 → (𝑧 = 𝑦 → ∀𝑥 𝑧 = 𝑦))
Theorem	nfeqf1 2414*	An equation between setvar is free of any other setvar. Usage of this theorem is discouraged because it depends on ax-13 2407. (Contributed by Wolf Lammen, 10-Jun-2019.) (New usage is discouraged.)
⊢ (¬ ∀𝑥 𝑥 = 𝑦 → Ⅎ𝑥 𝑦 = 𝑧)
Theorem	dveeq1 2415*	Quantifier introduction when one pair of variables is distinct. Usage of this theorem is discouraged because it depends on ax-13 2407. (Contributed by NM, 2-Jan-2002.) Remove dependency on ax-11 2195. (Revised by Wolf Lammen, 8-Sep-2018.) (New usage is discouraged.)
⊢ (¬ ∀𝑥 𝑥 = 𝑦 → (𝑦 = 𝑧 → ∀𝑥 𝑦 = 𝑧))
Theorem	nfeqf 2416	A variable is effectively not free in an equality if it is not either of the involved variables. Ⅎ version of ax-c9 39704. Usage of this theorem is discouraged because it depends on ax-13 2407. (Contributed by Mario Carneiro, 6-Oct-2016.) Remove dependency on ax-11 2195. (Revised by Wolf Lammen, 6-Sep-2018.) (New usage is discouraged.)
⊢ ((¬ ∀𝑧 𝑧 = 𝑥 ∧ ¬ ∀𝑧 𝑧 = 𝑦) → Ⅎ𝑧 𝑥 = 𝑦)
Theorem	axc9 2417	Derive set.mm's original ax-c9 39704 from the shorter ax-13 2407. Usage is discouraged to avoid uninformed ax-13 2407 propagation. (Contributed by NM, 29-Nov-2015.) (Revised by NM, 24-Dec-2015.) (Proof shortened by Wolf Lammen, 29-Apr-2018.) (New usage is discouraged.)
⊢ (¬ ∀𝑧 𝑧 = 𝑥 → (¬ ∀𝑧 𝑧 = 𝑦 → (𝑥 = 𝑦 → ∀𝑧 𝑥 = 𝑦)))
Theorem	ax6e 2418	At least one individual exists. This is not a theorem of free logic, which is sound in empty domains. For such a logic, we would add this theorem as an axiom of set theory (Axiom 0 of [Kunen] p. 10). In the system consisting of ax-4 1842 through ax-9 2156, all axioms other than ax-6 2000 are believed to be theorems of free logic, although the system without ax-6 2000 is not complete in free logic. Usage of this theorem is discouraged because it depends on ax-13 2407. It is preferred to use ax6ev 2002 when it is sufficient. (Contributed by NM, 14-May-1993.) Shortened after ax13lem1 2409 became available. (Revised by Wolf Lammen, 8-Sep-2018.) (New usage is discouraged.)
⊢ ∃𝑥 𝑥 = 𝑦
Theorem	ax6 2419	Theorem showing that ax-6 2000 follows from the weaker version ax6v 2001. (Even though this theorem depends on ax-6 2000, all references of ax-6 2000 are made via ax6v 2001. An earlier version stated ax6v 2001 as a separate axiom, but having two axioms caused some confusion.) This theorem should be referenced in place of ax-6 2000 so that all proofs can be traced back to ax6v 2001. When possible, use the weaker ax6v 2001 rather than ax6 2419 since the ax6v 2001 derivation is much shorter and requires fewer axioms. (Contributed by NM, 12-Nov-2013.) (Revised by NM, 25-Jul-2015.) (Proof shortened by Wolf Lammen, 4-Feb-2018.) Usage of this theorem is discouraged because it depends on ax-13 2407. Use ax6v 2001 instead. (New usage is discouraged.)
⊢ ¬ ∀𝑥 ¬ 𝑥 = 𝑦
Theorem	axc10 2420	Show that the original axiom ax-c10 39700 can be derived from ax6 2419 and axc7 2353 (on top of propositional calculus, ax-gen 1828, and ax-4 1842). See ax6fromc10 39710 for the rederivation of ax6 2419 from ax-c10 39700. Normally, axc10 2420 should be used rather than ax-c10 39700, except by theorems specifically studying the latter's properties. See bj-axc10v 37468 for a weaker version requiring fewer axioms. (Contributed by NM, 5-Aug-1993.) (Proof modification is discouraged.) Usage of this theorem is discouraged because it depends on ax-13 2407. (New usage is discouraged.)
⊢ (∀𝑥(𝑥 = 𝑦 → ∀𝑥𝜑) → 𝜑)
Theorem	spimt 2421	Closed theorem form of spim 2422. (Contributed by NM, 15-Jan-2008.) (Revised by Mario Carneiro, 17-Oct-2016.) (Proof shortened by Wolf Lammen, 21-Mar-2023.) Usage of this theorem is discouraged because it depends on ax-13 2407. (New usage is discouraged.)
⊢ ((Ⅎ𝑥𝜓 ∧ ∀𝑥(𝑥 = 𝑦 → (𝜑 → 𝜓))) → (∀𝑥𝜑 → 𝜓))
Theorem	spim 2422	Specialization, using implicit substitution. Compare Lemma 14 of [Tarski] p. 70. The spim 2422 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 2407. See spimw 2003 for a version requiring fewer 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.)
⊢ Ⅎ𝑥𝜓    &   ⊢ (𝑥 = 𝑦 → (𝜑 → 𝜓))    ⇒   ⊢ (∀𝑥𝜑 → 𝜓)
Theorem	spimed 2423	Deduction version of spime 2424. (Contributed by NM, 14-May-1993.) (Revised by Mario Carneiro, 3-Oct-2016.) (Proof shortened by Wolf Lammen, 19-Feb-2018.) Usage of this theorem is discouraged because it depends on ax-13 2407. Use spimedv 2236 instead. (New usage is discouraged.)
⊢ (𝜒 → Ⅎ𝑥𝜑)    &   ⊢ (𝑥 = 𝑦 → (𝜑 → 𝜓))    ⇒   ⊢ (𝜒 → (𝜑 → ∃𝑥𝜓))
Theorem	spime 2424	Existential introduction, using implicit substitution. Compare Lemma 14 of [Tarski] p. 70. See spimew 2004 and spimevw 2018 for weaker versions requiring fewer axioms. (Contributed by NM, 7-Aug-1994.) (Revised by Mario Carneiro, 3-Oct-2016.) (Proof shortened by Wolf Lammen, 6-Mar-2018.) Usage of this theorem is discouraged because it depends on ax-13 2407. Use spimefv 2237 instead. (New usage is discouraged.)
⊢ Ⅎ𝑥𝜑    &   ⊢ (𝑥 = 𝑦 → (𝜑 → 𝜓))    ⇒   ⊢ (𝜑 → ∃𝑥𝜓)
Theorem	spimv 2425*	A version of spim 2422 with a distinct variable requirement instead of a bound-variable hypothesis. See spimfv 2278 and spimvw 2019 for versions requiring fewer axioms. (Contributed by NM, 31-Jul-1993.) Usage of this theorem is discouraged because it depends on ax-13 2407. Use spimvw 2019 instead. (New usage is discouraged.)
⊢ (𝑥 = 𝑦 → (𝜑 → 𝜓))    ⇒   ⊢ (∀𝑥𝜑 → 𝜓)
Theorem	spimvALT 2426*	Alternate proof of spimv 2425. Note that it requires only ax-1 6 through ax-5 1943 together with ax6e 2418. Currently, proofs derive from ax6v 2001, but if ax-6 2000 could be used instead, this proof would reduce axiom usage. (Contributed by NM, 31-Jul-1993.) Remove dependency on ax-10 2179. (Revised by BJ, 29-Nov-2020.) (Proof modification is discouraged.) (New usage is discouraged.)
⊢ (𝑥 = 𝑦 → (𝜑 → 𝜓))    ⇒   ⊢ (∀𝑥𝜑 → 𝜓)
Theorem	spimev 2427*	Distinct-variable version of spime 2424. (Contributed by NM, 10-Jan-1993.) Usage of this theorem is discouraged because it depends on ax-13 2407. Use spimevw 2018 instead. (New usage is discouraged.)
⊢ (𝑥 = 𝑦 → (𝜑 → 𝜓))    ⇒   ⊢ (𝜑 → ∃𝑥𝜓)
Theorem	spv 2428*	Specialization, using implicit substitution. Usage of this theorem is discouraged because it depends on ax-13 2407. Use the weaker spvv 2021 if possible. (Contributed by NM, 30-Aug-1993.) (New usage is discouraged.)
⊢ (𝑥 = 𝑦 → (𝜑 ↔ 𝜓))    ⇒   ⊢ (∀𝑥𝜑 → 𝜓)
Theorem	spei 2429	Inference from existential specialization, using implicit substitution. Remove a distinct variable constraint. Usage of this theorem is discouraged because it depends on ax-13 2407. Use the weaker speiv 2005 if possible. (Contributed by NM, 19-Aug-1993.) (Proof shortened by Wolf Lammen, 12-May-2018.) (New usage is discouraged.)
⊢ (𝑥 = 𝑦 → (𝜑 ↔ 𝜓))    &   ⊢ 𝜓    ⇒   ⊢ ∃𝑥𝜑
Theorem	chvar 2430	Implicit substitution of 𝑦 for 𝑥 into a theorem. Usage of this theorem is discouraged because it depends on ax-13 2407. Use the weaker chvarfv 2279 if possible. (Contributed by Raph Levien, 9-Jul-2003.) (Revised by Mario Carneiro, 3-Oct-2016.) (New usage is discouraged.)
⊢ Ⅎ𝑥𝜓    &   ⊢ (𝑥 = 𝑦 → (𝜑 ↔ 𝜓))    &   ⊢ 𝜑    ⇒   ⊢ 𝜓
Theorem	chvarv 2431*	Implicit substitution of 𝑦 for 𝑥 into a theorem. Usage of this theorem is discouraged because it depends on ax-13 2407. Use the weaker chvarvv 2022 if possible. (Contributed by NM, 20-Apr-1994.) (Proof shortened by Wolf Lammen, 22-Apr-2018.) (New usage is discouraged.)
⊢ (𝑥 = 𝑦 → (𝜑 ↔ 𝜓))    &   ⊢ 𝜑    ⇒   ⊢ 𝜓
Theorem	cbv3 2432	Rule used to change bound variables, using implicit substitution, that does not use ax-c9 39704. Usage of this theorem is discouraged because it depends on ax-13 2407. Use the weaker cbv3v 2370 if possible. (Contributed by NM, 5-Aug-1993.) (Proof shortened by Wolf Lammen, 12-May-2018.) (New usage is discouraged.)
⊢ Ⅎ𝑦𝜑    &   ⊢ Ⅎ𝑥𝜓    &   ⊢ (𝑥 = 𝑦 → (𝜑 → 𝜓))    ⇒   ⊢ (∀𝑥𝜑 → ∀𝑦𝜓)
Theorem	cbval 2433	Rule used to change bound variables, using implicit substitution. Usage of this theorem is discouraged because it depends on ax-13 2407. Check out cbvalw 2068, cbvalvw 2069, cbvalv1 2376 for versions requiring fewer axioms. (Contributed by NM, 13-May-1993.) (Revised by Mario Carneiro, 3-Oct-2016.) (New usage is discouraged.)
⊢ Ⅎ𝑦𝜑    &   ⊢ Ⅎ𝑥𝜓    &   ⊢ (𝑥 = 𝑦 → (𝜑 ↔ 𝜓))    ⇒   ⊢ (∀𝑥𝜑 ↔ ∀𝑦𝜓)
Theorem	cbvex 2434	Rule used to change bound variables, using implicit substitution. Usage of this theorem is discouraged because it depends on ax-13 2407. Check out cbvexvw 2070, cbvexv1 2377 for weaker versions requiring fewer axioms. (Contributed by NM, 21-Jun-1993.) (New usage is discouraged.)
⊢ Ⅎ𝑦𝜑    &   ⊢ Ⅎ𝑥𝜓    &   ⊢ (𝑥 = 𝑦 → (𝜑 ↔ 𝜓))    ⇒   ⊢ (∃𝑥𝜑 ↔ ∃𝑦𝜓)
Theorem	cbvalv 2435*	Rule used to change bound variables, using implicit substitution. Usage of this theorem is discouraged because it depends on ax-13 2407. See cbvalvw 2069 for a version requiring fewer axioms, to be preferred when sufficient. (Contributed by NM, 5-Aug-1993.) Remove dependency on ax-10 2179 and shorten proof. (Revised by Wolf Lammen, 11-Sep-2023.) (New usage is discouraged.)
⊢ (𝑥 = 𝑦 → (𝜑 ↔ 𝜓))    ⇒   ⊢ (∀𝑥𝜑 ↔ ∀𝑦𝜓)
Theorem	cbvexv 2436*	Rule used to change bound variables, using implicit substitution. Usage of this theorem is discouraged because it depends on ax-13 2407. See cbvexvw 2070 for a version requiring fewer axioms, to be preferred when sufficient. (Contributed by NM, 21-Jun-1993.) Remove dependency on ax-10 2179 and shorten proof. (Revised by Wolf Lammen, 11-Sep-2023.) (New usage is discouraged.)
⊢ (𝑥 = 𝑦 → (𝜑 ↔ 𝜓))    ⇒   ⊢ (∃𝑥𝜑 ↔ ∃𝑦𝜓)
Theorem	cbv1 2437	Rule used to change bound variables, using implicit substitution. Usage of this theorem is discouraged because it depends on ax-13 2407. See cbv1v 2371 with disjoint variable conditions, not depending on ax-13 2407. (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.)
⊢ Ⅎ𝑥𝜑    &   ⊢ Ⅎ𝑦𝜑    &   ⊢ (𝜑 → Ⅎ𝑦𝜓)    &   ⊢ (𝜑 → Ⅎ𝑥𝜒)    &   ⊢ (𝜑 → (𝑥 = 𝑦 → (𝜓 → 𝜒)))    ⇒   ⊢ (𝜑 → (∀𝑥𝜓 → ∀𝑦𝜒))
Theorem	cbv2 2438	Rule used to change bound variables, using implicit substitution. Usage of this theorem is discouraged because it depends on ax-13 2407. See cbv2w 2372 with disjoint variable conditions, not depending on ax-13 2407. (Contributed by NM, 5-Aug-1993.) (Revised by Mario Carneiro, 3-Oct-2016.) Format hypotheses to common style, avoid ax-10 2179. (Revised by Wolf Lammen, 10-Sep-2023.) (New usage is discouraged.)
⊢ Ⅎ𝑥𝜑    &   ⊢ Ⅎ𝑦𝜑    &   ⊢ (𝜑 → Ⅎ𝑦𝜓)    &   ⊢ (𝜑 → Ⅎ𝑥𝜒)    &   ⊢ (𝜑 → (𝑥 = 𝑦 → (𝜓 ↔ 𝜒)))    ⇒   ⊢ (𝜑 → (∀𝑥𝜓 ↔ ∀𝑦𝜒))
Theorem	cbv3h 2439	Rule used to change bound variables, using implicit substitution. Usage of this theorem is discouraged because it depends on ax-13 2407. Use the weaker cbv3hv 2375 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 → (𝜑 → 𝜓))    ⇒   ⊢ (∀𝑥𝜑 → ∀𝑦𝜓)
Theorem	cbv1h 2440	Rule used to change bound variables, using implicit substitution. Usage of this theorem is discouraged because it depends on ax-13 2407. (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 → (𝜓 → 𝜒)))    ⇒   ⊢ (∀𝑥∀𝑦𝜑 → (∀𝑥𝜓 → ∀𝑦𝜒))
Theorem	cbv2h 2441	Rule used to change bound variables, using implicit substitution. Usage of this theorem is discouraged because it depends on ax-13 2407. (Contributed by NM, 11-May-1993.) (New usage is discouraged.)
⊢ (𝜑 → (𝜓 → ∀𝑦𝜓))    &   ⊢ (𝜑 → (𝜒 → ∀𝑥𝜒))    &   ⊢ (𝜑 → (𝑥 = 𝑦 → (𝜓 ↔ 𝜒)))    ⇒   ⊢ (∀𝑥∀𝑦𝜑 → (∀𝑥𝜓 ↔ ∀𝑦𝜒))
Theorem	cbvald 2442*	Deduction used to change bound variables, using implicit substitution, particularly useful in conjunction with dvelim 2486. Usage of this theorem is discouraged because it depends on ax-13 2407. See cbvaldw 2373 for a version with 𝑥, 𝑦 disjoint, not depending on ax-13 2407. (Contributed by NM, 2-Jan-2002.) (Revised by Mario Carneiro, 6-Oct-2016.) (Revised by Wolf Lammen, 13-May-2018.) (New usage is discouraged.)
⊢ Ⅎ𝑦𝜑    &   ⊢ (𝜑 → Ⅎ𝑦𝜓)    &   ⊢ (𝜑 → (𝑥 = 𝑦 → (𝜓 ↔ 𝜒)))    ⇒   ⊢ (𝜑 → (∀𝑥𝜓 ↔ ∀𝑦𝜒))
Theorem	cbvexd 2443*	Deduction used to change bound variables, using implicit substitution, particularly useful in conjunction with dvelim 2486. Usage of this theorem is discouraged because it depends on ax-13 2407. Use the weaker cbvexdw 2374 if possible. (Contributed by NM, 2-Jan-2002.) (Revised by Mario Carneiro, 6-Oct-2016.) (New usage is discouraged.)
⊢ Ⅎ𝑦𝜑    &   ⊢ (𝜑 → Ⅎ𝑦𝜓)    &   ⊢ (𝜑 → (𝑥 = 𝑦 → (𝜓 ↔ 𝜒)))    ⇒   ⊢ (𝜑 → (∃𝑥𝜓 ↔ ∃𝑦𝜒))
Theorem	cbvaldva 2444*	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 2407. Use the weaker cbvaldvaw 2071 if possible. (Contributed by David Moews, 1-May-2017.) (New usage is discouraged.)
⊢ ((𝜑 ∧ 𝑥 = 𝑦) → (𝜓 ↔ 𝜒))    ⇒   ⊢ (𝜑 → (∀𝑥𝜓 ↔ ∀𝑦𝜒))
Theorem	cbvexdva 2445*	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 2407. Use the weaker cbvexdvaw 2072 if possible. (Contributed by David Moews, 1-May-2017.) (New usage is discouraged.)
⊢ ((𝜑 ∧ 𝑥 = 𝑦) → (𝜓 ↔ 𝜒))    ⇒   ⊢ (𝜑 → (∃𝑥𝜓 ↔ ∃𝑦𝜒))
Theorem	cbval2 2446*	Rule used to change bound variables, using implicit substitution. Usage of this theorem is discouraged because it depends on ax-13 2407. Use the weaker cbval2v 2378 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.)
⊢ Ⅎ𝑧𝜑    &   ⊢ Ⅎ𝑤𝜑    &   ⊢ Ⅎ𝑥𝜓    &   ⊢ Ⅎ𝑦𝜓    &   ⊢ ((𝑥 = 𝑧 ∧ 𝑦 = 𝑤) → (𝜑 ↔ 𝜓))    ⇒   ⊢ (∀𝑥∀𝑦𝜑 ↔ ∀𝑧∀𝑤𝜓)
Theorem	cbvex2 2447*	Rule used to change bound variables, using implicit substitution. Usage of this theorem is discouraged because it depends on ax-13 2407. Use the weaker cbvex2v 2379 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.)
⊢ Ⅎ𝑧𝜑    &   ⊢ Ⅎ𝑤𝜑    &   ⊢ Ⅎ𝑥𝜓    &   ⊢ Ⅎ𝑦𝜓    &   ⊢ ((𝑥 = 𝑧 ∧ 𝑦 = 𝑤) → (𝜑 ↔ 𝜓))    ⇒   ⊢ (∃𝑥∃𝑦𝜑 ↔ ∃𝑧∃𝑤𝜓)
Theorem	cbval2vv 2448*	Rule used to change bound variables, using implicit substitution. Usage of this theorem is discouraged because it depends on ax-13 2407. Use the weaker cbval2vw 2073 if possible. (Contributed by NM, 4-Feb-2005.) Remove dependency on ax-10 2179. (Revised by Wolf Lammen, 18-Jul-2021.) (New usage is discouraged.)
⊢ ((𝑥 = 𝑧 ∧ 𝑦 = 𝑤) → (𝜑 ↔ 𝜓))    ⇒   ⊢ (∀𝑥∀𝑦𝜑 ↔ ∀𝑧∀𝑤𝜓)
Theorem	cbvex2vv 2449*	Rule used to change bound variables, using implicit substitution. Usage of this theorem is discouraged because it depends on ax-13 2407. Use the weaker cbvex2vw 2074 if possible. (Contributed by NM, 26-Jul-1995.) Remove dependency on ax-10 2179. (Revised by Wolf Lammen, 18-Jul-2021.) (New usage is discouraged.)
⊢ ((𝑥 = 𝑧 ∧ 𝑦 = 𝑤) → (𝜑 ↔ 𝜓))    ⇒   ⊢ (∃𝑥∃𝑦𝜑 ↔ ∃𝑧∃𝑤𝜓)
Theorem	cbvex4v 2450*	Rule used to change bound variables, using implicit substitution. Usage of this theorem is discouraged because it depends on ax-13 2407. Use the weaker cbvex4vw 2075 if possible. (Contributed by NM, 26-Jul-1995.) (New usage is discouraged.)
⊢ ((𝑥 = 𝑣 ∧ 𝑦 = 𝑢) → (𝜑 ↔ 𝜓))    &   ⊢ ((𝑧 = 𝑓 ∧ 𝑤 = 𝑔) → (𝜓 ↔ 𝜒))    ⇒   ⊢ (∃𝑥∃𝑦∃𝑧∃𝑤𝜑 ↔ ∃𝑣∃𝑢∃𝑓∃𝑔𝜒)
Theorem	equs4 2451	Lemma used in proofs of implicit substitution properties. The converse requires either a disjoint variable condition (sbalex 2281) or a nonfreeness hypothesis (equs45f 2494). Usage of this theorem is discouraged because it depends on ax-13 2407. See equs4v 2033 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 = 𝑦 → 𝜑) → ∃𝑥(𝑥 = 𝑦 ∧ 𝜑))
Theorem	equsal 2452	An equivalence related to implicit substitution. Usage of this theorem is discouraged because it depends on ax-13 2407. See equsalvw 2037 and equsalv 2306 for versions with disjoint variable conditions proved from fewer axioms. See also the dual form equsex 2453. (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.)
⊢ Ⅎ𝑥𝜓    &   ⊢ (𝑥 = 𝑦 → (𝜑 ↔ 𝜓))    ⇒   ⊢ (∀𝑥(𝑥 = 𝑦 → 𝜑) ↔ 𝜓)
Theorem	equsex 2453	An equivalence related to implicit substitution. Usage of this theorem is discouraged because it depends on ax-13 2407. See equsexvw 2038 and equsexv 2307 for versions with disjoint variable conditions proved from fewer axioms. See also the dual form equsal 2452. See equsexALT 2454 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.)
⊢ Ⅎ𝑥𝜓    &   ⊢ (𝑥 = 𝑦 → (𝜑 ↔ 𝜓))    ⇒   ⊢ (∃𝑥(𝑥 = 𝑦 ∧ 𝜑) ↔ 𝜓)
Theorem	equsexALT 2454	Alternate proof of equsex 2453. This proves the result directly, instead of as a corollary of equsal 2452 via equs4 2451. Note in particular that only existential quantifiers appear in the proof and that the only step requiring ax-13 2407 is ax6e 2418. This proof mimics that of equsal 2452 (in particular, note that pm5.32i 585, exbii 1881, 19.41 2274, mpbiran 722 correspond respectively to pm5.74i 274, albii 1852, 19.23 2250, a1bi 365). (Contributed by BJ, 20-Aug-2020.) (Proof modification is discouraged.) (New usage is discouraged.)
⊢ Ⅎ𝑥𝜓    &   ⊢ (𝑥 = 𝑦 → (𝜑 ↔ 𝜓))    ⇒   ⊢ (∃𝑥(𝑥 = 𝑦 ∧ 𝜑) ↔ 𝜓)
Theorem	equsalh 2455	An equivalence related to implicit substitution. Usage of this theorem is discouraged because it depends on ax-13 2407. See equsalhw 2329 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 → (𝜑 ↔ 𝜓))    ⇒   ⊢ (∀𝑥(𝑥 = 𝑦 → 𝜑) ↔ 𝜓)
Theorem	equsexh 2456	An equivalence related to implicit substitution. Usage of this theorem is discouraged because it depends on ax-13 2407. See equsexhv 2330 for a version with a disjoint variable condition which does not require ax-13 2407. (Contributed by NM, 5-Aug-1993.) (New usage is discouraged.)
⊢ (𝜓 → ∀𝑥𝜓)    &   ⊢ (𝑥 = 𝑦 → (𝜑 ↔ 𝜓))    ⇒   ⊢ (∃𝑥(𝑥 = 𝑦 ∧ 𝜑) ↔ 𝜓)
Theorem	axc15 2457	Derivation of set.mm's original ax-c15 39703 from ax-c11n 39702 and the shorter ax-12 2216 that has replaced it. Theorem ax12 2458 shows the reverse derivation of ax-12 2216 from ax-c15 39703. Normally, axc15 2457 should be used rather than ax-c15 39703, except by theorems specifically studying the latter's properties. Usage of this theorem is discouraged because it depends on ax-13 2407. (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 = 𝑦 → 𝜑))))
Theorem	ax12 2458	Rederivation of Axiom ax-12 2216 from ax12v 2217 (used only via sp 2222), axc11r 2403, and axc15 2457 (on top of Tarski's FOL). Since this version depends on ax-13 2407, usage of the weaker ax12v 2217, ax12w 2171, ax12i 1999 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 = 𝑦 → 𝜑)))
Theorem	ax12b 2459	A bidirectional version of axc15 2457. Usage of this theorem is discouraged because it depends on ax-13 2407. (Contributed by NM, 30-Jun-2006.) (New usage is discouraged.)
⊢ ((¬ ∀𝑥 𝑥 = 𝑦 ∧ 𝑥 = 𝑦) → (𝜑 ↔ ∀𝑥(𝑥 = 𝑦 → 𝜑)))
Theorem	ax13ALT 2460	Alternate proof of ax13 2410 from FOL, sp 2222, and axc9 2417. (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))
Theorem	axc11n 2461	Derive set.mm's original ax-c11n 39702 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 2090. Use aecom 2462 instead when this does not lengthen the proof. Usage of this theorem is discouraged because it depends on ax-13 2407. (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)
Theorem	aecom 2462	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 2407. (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)
Theorem	aecoms 2463	A commutation rule for identical variable specifiers. Usage of this theorem is discouraged because it depends on ax-13 2407. (Contributed by NM, 10-May-1993.) (New usage is discouraged.)
⊢ (∀𝑥 𝑥 = 𝑦 → 𝜑)    ⇒   ⊢ (∀𝑦 𝑦 = 𝑥 → 𝜑)
Theorem	naecoms 2464	A commutation rule for distinct variable specifiers. Usage of this theorem is discouraged because it depends on ax-13 2407. (Contributed by NM, 2-Jan-2002.) (New usage is discouraged.)
⊢ (¬ ∀𝑥 𝑥 = 𝑦 → 𝜑)    ⇒   ⊢ (¬ ∀𝑦 𝑦 = 𝑥 → 𝜑)
Theorem	axc11 2465	Show that ax-c11 39701 can be derived from ax-c11n 39702 in the form of axc11n 2461. Normally, axc11 2465 should be used rather than ax-c11 39701, except by theorems specifically studying the latter's properties. Usage of this theorem is discouraged because it depends on ax-13 2407. Use the weaker axc11v 2303 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))
Theorem	hbae 2466	All variables are effectively bound in an identical variable specifier. Usage of this theorem is discouraged because it depends on ax-13 2407. Use the weaker hbaev 2094 when possible. (Contributed by NM, 13-May-1993.) (Proof shortened by Wolf Lammen, 21-Apr-2018.) (New usage is discouraged.)
⊢ (∀𝑥 𝑥 = 𝑦 → ∀𝑧∀𝑥 𝑥 = 𝑦)
Theorem	hbnae 2467	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 2407. Use the weaker hbnaev 2097 when possible. (Contributed by NM, 13-May-1993.) (New usage is discouraged.)
⊢ (¬ ∀𝑥 𝑥 = 𝑦 → ∀𝑧 ¬ ∀𝑥 𝑥 = 𝑦)
Theorem	nfae 2468	All variables are effectively bound in an identical variable specifier. Usage of this theorem is discouraged because it depends on ax-13 2407. (Contributed by Mario Carneiro, 11-Aug-2016.) (New usage is discouraged.)
⊢ Ⅎ𝑧∀𝑥 𝑥 = 𝑦
Theorem	nfnae 2469	All variables are effectively bound in a distinct variable specifier. Usage of this theorem is discouraged because it depends on ax-13 2407. Use the weaker nfnaew 2187 when possible. (Contributed by Mario Carneiro, 11-Aug-2016.) (New usage is discouraged.)
⊢ Ⅎ𝑧 ¬ ∀𝑥 𝑥 = 𝑦
Theorem	hbnaes 2470	Rule that applies hbnae 2467 to antecedent. Usage of this theorem is discouraged because it depends on ax-13 2407. (Contributed by NM, 15-May-1993.) (New usage is discouraged.)
⊢ (∀𝑧 ¬ ∀𝑥 𝑥 = 𝑦 → 𝜑)    ⇒   ⊢ (¬ ∀𝑥 𝑥 = 𝑦 → 𝜑)
Theorem	axc16i 2471*	Inference with axc16 2300 as its conclusion. (Contributed by NM, 20-May-2008.) (Proof modification is discouraged.) Usage of this theorem is discouraged because it depends on ax-13 2407. Use axc16 2300 instead. (New usage is discouraged.)
⊢ (𝑥 = 𝑧 → (𝜑 ↔ 𝜓))    &   ⊢ (𝜓 → ∀𝑥𝜓)    ⇒   ⊢ (∀𝑥 𝑥 = 𝑦 → (𝜑 → ∀𝑥𝜑))
Theorem	axc16nfALT 2472*	Alternate proof of axc16nf 2302, shorter but requiring ax-11 2195 and ax-13 2407. (Contributed by Mario Carneiro, 7-Oct-2016.) (Proof modification is discouraged.) (New usage is discouraged.)
⊢ (∀𝑥 𝑥 = 𝑦 → Ⅎ𝑧𝜑)
Theorem	dral2 2473	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 2407. Usage of albidv 1953 is preferred, which requires fewer axioms. (Contributed by NM, 27-Feb-2005.) Allow a shortening of dral1 2474. (Revised by Wolf Lammen, 4-Mar-2018.) (New usage is discouraged.)
⊢ (∀𝑥 𝑥 = 𝑦 → (𝜑 ↔ 𝜓))    ⇒   ⊢ (∀𝑥 𝑥 = 𝑦 → (∀𝑧𝜑 ↔ ∀𝑧𝜓))
Theorem	dral1 2474	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 2407. Use the weaker dral1v 2404 if possible. (Contributed by NM, 24-Nov-1994.) Remove dependency on ax-11 2195. (Revised by Wolf Lammen, 6-Sep-2018.) (New usage is discouraged.)
⊢ (∀𝑥 𝑥 = 𝑦 → (𝜑 ↔ 𝜓))    ⇒   ⊢ (∀𝑥 𝑥 = 𝑦 → (∀𝑥𝜑 ↔ ∀𝑦𝜓))
Theorem	dral1ALT 2475	Alternate proof of dral1 2474, shorter but requiring ax-11 2195. (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 → (𝜑 ↔ 𝜓))    ⇒   ⊢ (∀𝑥 𝑥 = 𝑦 → (∀𝑥𝜑 ↔ ∀𝑦𝜓))
Theorem	drex1 2476	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 2407. Use the weaker drex1v 2405 if possible. (Contributed by NM, 27-Feb-2005.) (New usage is discouraged.)
⊢ (∀𝑥 𝑥 = 𝑦 → (𝜑 ↔ 𝜓))    ⇒   ⊢ (∀𝑥 𝑥 = 𝑦 → (∃𝑥𝜑 ↔ ∃𝑦𝜓))
Theorem	drex2 2477	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 2407. Usage of exbidv 1954 is preferred, which requires fewer axioms. (Contributed by NM, 27-Feb-2005.) (New usage is discouraged.)
⊢ (∀𝑥 𝑥 = 𝑦 → (𝜑 ↔ 𝜓))    ⇒   ⊢ (∀𝑥 𝑥 = 𝑦 → (∃𝑧𝜑 ↔ ∃𝑧𝜓))
Theorem	drnf1 2478	Formula-building lemma for use with the Distinctor Reduction Theorem. (Contributed by Mario Carneiro, 4-Oct-2016.) Usage of this theorem is discouraged because it depends on ax-13 2407. Use drnf1v 2406 instead. (New usage is discouraged.)
⊢ (∀𝑥 𝑥 = 𝑦 → (𝜑 ↔ 𝜓))    ⇒   ⊢ (∀𝑥 𝑥 = 𝑦 → (Ⅎ𝑥𝜑 ↔ Ⅎ𝑦𝜓))
Theorem	drnf2 2479	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.) Usage of this theorem is discouraged because it depends on ax-13 2407. Use nfbidv 1955 instead. (New usage is discouraged.)
⊢ (∀𝑥 𝑥 = 𝑦 → (𝜑 ↔ 𝜓))    ⇒   ⊢ (∀𝑥 𝑥 = 𝑦 → (Ⅎ𝑧𝜑 ↔ Ⅎ𝑧𝜓))
Theorem	nfald2 2480	Variation on nfald 2364 which adds the hypothesis that 𝑥 and 𝑦 are distinct in the inner subproof. (Contributed by Mario Carneiro, 8-Oct-2016.) Usage of this theorem is discouraged because it depends on ax-13 2407. Use nfald 2364 instead. (New usage is discouraged.)
⊢ Ⅎ𝑦𝜑    &   ⊢ ((𝜑 ∧ ¬ ∀𝑥 𝑥 = 𝑦) → Ⅎ𝑥𝜓)    ⇒   ⊢ (𝜑 → Ⅎ𝑥∀𝑦𝜓)
Theorem	nfexd2 2481	Variation on nfexd 2365 which adds the hypothesis that 𝑥 and 𝑦 are distinct in the inner subproof. (Contributed by Mario Carneiro, 8-Oct-2016.) Usage of this theorem is discouraged because it depends on ax-13 2407. Use nfexd 2365 instead. (New usage is discouraged.)
⊢ Ⅎ𝑦𝜑    &   ⊢ ((𝜑 ∧ ¬ ∀𝑥 𝑥 = 𝑦) → Ⅎ𝑥𝜓)    ⇒   ⊢ (𝜑 → Ⅎ𝑥∃𝑦𝜓)
Theorem	exdistrf 2482	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.) Usage of this theorem is discouraged because it depends on ax-13 2407. Use exdistr 1987 instead. (New usage is discouraged.)
⊢ (¬ ∀𝑥 𝑥 = 𝑦 → Ⅎ𝑦𝜑)    ⇒   ⊢ (∃𝑥∃𝑦(𝜑 ∧ 𝜓) → ∃𝑥(𝜑 ∧ ∃𝑦𝜓))
Theorem	dvelimf 2483	Version of dvelimv 2487 without any variable restrictions. Usage of this theorem is discouraged because it depends on ax-13 2407. (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.)
⊢ Ⅎ𝑥𝜑    &   ⊢ Ⅎ𝑧𝜓    &   ⊢ (𝑧 = 𝑦 → (𝜑 ↔ 𝜓))    ⇒   ⊢ (¬ ∀𝑥 𝑥 = 𝑦 → Ⅎ𝑥𝜓)
Theorem	dvelimdf 2484	Deduction form of dvelimf 2483. Usage of this theorem is discouraged because it depends on ax-13 2407. (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.)
⊢ Ⅎ𝑥𝜑    &   ⊢ Ⅎ𝑧𝜑    &   ⊢ (𝜑 → Ⅎ𝑥𝜓)    &   ⊢ (𝜑 → Ⅎ𝑧𝜒)    &   ⊢ (𝜑 → (𝑧 = 𝑦 → (𝜓 ↔ 𝜒)))    ⇒   ⊢ (𝜑 → (¬ ∀𝑥 𝑥 = 𝑦 → Ⅎ𝑥𝜒))
Theorem	dvelimh 2485	Version of dvelim 2486 without any variable restrictions. Usage of this theorem is discouraged because it depends on ax-13 2407. Check out dvelimhw 2380 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 → (𝜑 ↔ 𝜓))    ⇒   ⊢ (¬ ∀𝑥 𝑥 = 𝑦 → (𝜓 → ∀𝑥𝜓))
Theorem	dvelim 2486*	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 ∀𝑥∀𝑧, conjoin them, and apply dvelimdf 2484. Other variants of this theorem are dvelimh 2485 (with no distinct variable restrictions) and dvelimhw 2380 (that avoids ax-13 2407). Usage of this theorem is discouraged because it depends on ax-13 2407. Check out dvelimhw 2380 for a version requiring fewer axioms. (Contributed by NM, 23-Nov-1994.) (New usage is discouraged.)
⊢ (𝜑 → ∀𝑥𝜑)    &   ⊢ (𝑧 = 𝑦 → (𝜑 ↔ 𝜓))    ⇒   ⊢ (¬ ∀𝑥 𝑥 = 𝑦 → (𝜓 → ∀𝑥𝜓))
Theorem	dvelimv 2487*	Similar to dvelim 2486 with first hypothesis replaced by a distinct variable condition. Usage of this theorem is discouraged because it depends on ax-13 2407. Check out dvelimhw 2380 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 → (𝜑 ↔ 𝜓))    ⇒   ⊢ (¬ ∀𝑥 𝑥 = 𝑦 → (𝜓 → ∀𝑥𝜓))
Theorem	dvelimnf 2488*	Version of dvelim 2486 using "not free" notation. Usage of this theorem is discouraged because it depends on ax-13 2407. (Contributed by Mario Carneiro, 9-Oct-2016.) (New usage is discouraged.)
⊢ Ⅎ𝑥𝜑    &   ⊢ (𝑧 = 𝑦 → (𝜑 ↔ 𝜓))    ⇒   ⊢ (¬ ∀𝑥 𝑥 = 𝑦 → Ⅎ𝑥𝜓)
Theorem	dveeq2ALT 2489*	Alternate proof of dveeq2 2413, shorter but requiring ax-11 2195. (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))
Theorem	equvini 2490	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 2407. See equvinv 2062 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 = 𝑧 ∧ 𝑧 = 𝑦))
Theorem	equvel 2491	A variable elimination law for equality with no distinct variable requirements. Compare equvini 2490. Usage of this theorem is discouraged because it depends on ax-13 2407. Use equvelv 2064 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 = 𝑥 ↔ 𝑧 = 𝑦) → 𝑥 = 𝑦)
Theorem	equs5a 2492	A property related to substitution that unlike equs5 2495 does not require a distinctor antecedent. Usage of this theorem is discouraged because it depends on ax-13 2407. This proof uses ax12 2458, see equs5aALT 2401 for an alternative one using ax-12 2216 but not ax13 2410. Usage of the weaker equs5av 2315 is preferred, which uses ax12v2 2218, but not ax-13 2407. (Contributed by NM, 2-Feb-2007.) (New usage is discouraged.)
⊢ (∃𝑥(𝑥 = 𝑦 ∧ ∀𝑦𝜑) → ∀𝑥(𝑥 = 𝑦 → 𝜑))
Theorem	equs5e 2493	A property related to substitution that unlike equs5 2495 does not require a distinctor antecedent. This proof uses ax12 2458, see equs5eALT 2402 for an alternative one using ax-12 2216 but not ax13 2410. Usage of this theorem is discouraged because it depends on ax-13 2407. (Contributed by NM, 2-Feb-2007.) (Proof shortened by Wolf Lammen, 15-Jan-2018.) (New usage is discouraged.)
⊢ (∃𝑥(𝑥 = 𝑦 ∧ 𝜑) → ∀𝑥(𝑥 = 𝑦 → ∃𝑦𝜑))
Theorem	equs45f 2494	Two ways of expressing substitution when 𝑦 is not free in 𝜑. The implication "to the left" is equs4 2451 and does not require the nonfreeness hypothesis. Theorem sbalex 2281 replaces the nonfreeness hypothesis with a disjoint variable condition and equs5 2495 replaces it with a distinctor antecedent. (Contributed by NM, 25-Apr-2008.) (Revised by Mario Carneiro, 4-Oct-2016.) Usage of this theorem is discouraged because it depends on ax-13 2407. Use sbalex 2281 instead. (New usage is discouraged.)
⊢ Ⅎ𝑦𝜑    ⇒   ⊢ (∃𝑥(𝑥 = 𝑦 ∧ 𝜑) ↔ ∀𝑥(𝑥 = 𝑦 → 𝜑))
Theorem	equs5 2495	Lemma used in proofs of substitution properties. If there is a disjoint variable condition on 𝑥, 𝑦, then sbalex 2281 can be used instead; if 𝑦 is not free in 𝜑, then equs45f 2494 can be used. Usage of this theorem is discouraged because it depends on ax-13 2407. (Contributed by NM, 14-May-1993.) (Revised by BJ, 1-Oct-2018.) (New usage is discouraged.)
⊢ (¬ ∀𝑥 𝑥 = 𝑦 → (∃𝑥(𝑥 = 𝑦 ∧ 𝜑) ↔ ∀𝑥(𝑥 = 𝑦 → 𝜑)))
Theorem	dveel1 2496*	Quantifier introduction when one pair of variables is disjoint. Usage of this theorem is discouraged because it depends on ax-13 2407. (Contributed by NM, 2-Jan-2002.) (New usage is discouraged.)
⊢ (¬ ∀𝑥 𝑥 = 𝑦 → (𝑦 ∈ 𝑧 → ∀𝑥 𝑦 ∈ 𝑧))
Theorem	dveel2 2497*	Quantifier introduction when one pair of variables is disjoint. Usage of this theorem is discouraged because it depends on ax-13 2407. (Contributed by NM, 2-Jan-2002.) (New usage is discouraged.)
⊢ (¬ ∀𝑥 𝑥 = 𝑦 → (𝑧 ∈ 𝑦 → ∀𝑥 𝑧 ∈ 𝑦))
Theorem	axc14 2498	Axiom ax-c14 39705 is redundant if we assume ax-5 1943. 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 2497 and ax-5 1943. 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 2407. (Contributed by NM, 29-Jun-1995.) (Proof modification is discouraged.) (New usage is discouraged.)
⊢ (¬ ∀𝑧 𝑧 = 𝑥 → (¬ ∀𝑧 𝑧 = 𝑦 → (𝑥 ∈ 𝑦 → ∀𝑧 𝑥 ∈ 𝑦)))
Theorem	sb6x 2499	Equivalence involving substitution for a variable not free. Usage of this theorem is discouraged because it depends on ax-13 2407. Usage of sb6 2122 is preferred, which requires fewer axioms. (Contributed by NM, 2-Jun-1993.) (Revised by Mario Carneiro, 4-Oct-2016.) (New usage is discouraged.)
⊢ Ⅎ𝑥𝜑    ⇒   ⊢ ([𝑦 / 𝑥]𝜑 ↔ ∀𝑥(𝑥 = 𝑦 → 𝜑))
Theorem	sbequ5 2500	Substitution does not change an identical variable specifier. Usage of this theorem is discouraged because it depends on ax-13 2407. (Contributed by NM, 15-May-1993.) (New usage is discouraged.)
⊢ ([𝑤 / 𝑧]∀𝑥 𝑥 = 𝑦 ↔ ∀𝑥 𝑥 = 𝑦)