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	cbvex 2401	Rule used to change bound variables, using implicit substitution. Usage of this theorem is discouraged because it depends on ax-13 2374. Check out cbvexvw 2044, cbvexv1 2343 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 2402*	Rule used to change bound variables, using implicit substitution. Usage of this theorem is discouraged because it depends on ax-13 2374. See cbvalvw 2043 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 → (𝜑 ↔ 𝜓))    ⇒   ⊢ (∀𝑥𝜑 ↔ ∀𝑦𝜓)
Theorem	cbvexv 2403*	Rule used to change bound variables, using implicit substitution. Usage of this theorem is discouraged because it depends on ax-13 2374. See cbvexvw 2044 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 → (𝜑 ↔ 𝜓))    ⇒   ⊢ (∃𝑥𝜑 ↔ ∃𝑦𝜓)
Theorem	cbv1 2404	Rule used to change bound variables, using implicit substitution. Usage of this theorem is discouraged because it depends on ax-13 2374. See cbv1v 2337 with disjoint variable conditions, not depending on ax-13 2374. (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 2405	Rule used to change bound variables, using implicit substitution. Usage of this theorem is discouraged because it depends on ax-13 2374. See cbv2w 2338 with disjoint variable conditions, not depending on ax-13 2374. (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.)
⊢ Ⅎ𝑥𝜑    &   ⊢ Ⅎ𝑦𝜑    &   ⊢ (𝜑 → Ⅎ𝑦𝜓)    &   ⊢ (𝜑 → Ⅎ𝑥𝜒)    &   ⊢ (𝜑 → (𝑥 = 𝑦 → (𝜓 ↔ 𝜒)))    ⇒   ⊢ (𝜑 → (∀𝑥𝜓 ↔ ∀𝑦𝜒))
Theorem	cbv3h 2406	Rule used to change bound variables, using implicit substitution. Usage of this theorem is discouraged because it depends on ax-13 2374. Use the weaker cbv3hv 2341 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 2407	Rule used to change bound variables, using implicit substitution. Usage of this theorem is discouraged because it depends on ax-13 2374. (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 2408	Rule used to change bound variables, using implicit substitution. Usage of this theorem is discouraged because it depends on ax-13 2374. (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 2409*	Deduction used to change bound variables, using implicit substitution, particularly useful in conjunction with dvelim 2453. Usage of this theorem is discouraged because it depends on ax-13 2374. See cbvaldw 2339 for a version with 𝑥, 𝑦 disjoint, not depending on ax-13 2374. (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 2410*	Deduction used to change bound variables, using implicit substitution, particularly useful in conjunction with dvelim 2453. Usage of this theorem is discouraged because it depends on ax-13 2374. Use the weaker cbvexdw 2340 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 2411*	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 2374. Use the weaker cbvaldvaw 2045 if possible. (Contributed by David Moews, 1-May-2017.) (New usage is discouraged.)
⊢ ((𝜑 ∧ 𝑥 = 𝑦) → (𝜓 ↔ 𝜒))    ⇒   ⊢ (𝜑 → (∀𝑥𝜓 ↔ ∀𝑦𝜒))
Theorem	cbvexdva 2412*	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 2374. Use the weaker cbvexdvaw 2046 if possible. (Contributed by David Moews, 1-May-2017.) (New usage is discouraged.)
⊢ ((𝜑 ∧ 𝑥 = 𝑦) → (𝜓 ↔ 𝜒))    ⇒   ⊢ (𝜑 → (∃𝑥𝜓 ↔ ∃𝑦𝜒))
Theorem	cbval2 2413*	Rule used to change bound variables, using implicit substitution. Usage of this theorem is discouraged because it depends on ax-13 2374. Use the weaker cbval2v 2344 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 2414*	Rule used to change bound variables, using implicit substitution. Usage of this theorem is discouraged because it depends on ax-13 2374. Use the weaker cbvex2v 2346 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 2415*	Rule used to change bound variables, using implicit substitution. Usage of this theorem is discouraged because it depends on ax-13 2374. Use the weaker cbval2vw 2047 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 = 𝑧 ∧ 𝑦 = 𝑤) → (𝜑 ↔ 𝜓))    ⇒   ⊢ (∀𝑥∀𝑦𝜑 ↔ ∀𝑧∀𝑤𝜓)
Theorem	cbvex2vv 2416*	Rule used to change bound variables, using implicit substitution. Usage of this theorem is discouraged because it depends on ax-13 2374. Use the weaker cbvex2vw 2048 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 = 𝑧 ∧ 𝑦 = 𝑤) → (𝜑 ↔ 𝜓))    ⇒   ⊢ (∃𝑥∃𝑦𝜑 ↔ ∃𝑧∃𝑤𝜓)
Theorem	cbvex4v 2417*	Rule used to change bound variables, using implicit substitution. Usage of this theorem is discouraged because it depends on ax-13 2374. Use the weaker cbvex4vw 2049 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 2418	Lemma used in proofs of implicit substitution properties. The converse requires either a disjoint variable condition (sbalex 2239) or a nonfreeness hypothesis (equs45f 2461). Usage of this theorem is discouraged because it depends on ax-13 2374. See equs4v 2007 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 2419	An equivalence related to implicit substitution. Usage of this theorem is discouraged because it depends on ax-13 2374. See equsalvw 2011 and equsalv 2263 for versions with disjoint variable conditions proved from fewer axioms. See also the dual form equsex 2420. (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 2420	An equivalence related to implicit substitution. Usage of this theorem is discouraged because it depends on ax-13 2374. See equsexvw 2012 and equsexv 2264 for versions with disjoint variable conditions proved from fewer axioms. See also the dual form equsal 2419. See equsexALT 2421 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 2421	Alternate proof of equsex 2420. This proves the result directly, instead of as a corollary of equsal 2419 via equs4 2418. Note in particular that only existential quantifiers appear in the proof and that the only step requiring ax-13 2374 is ax6e 2385. This proof mimics that of equsal 2419 (in particular, note that pm5.32i 575, exbii 1854, 19.41 2232, mpbiran 706 correspond respectively to pm5.74i 270, albii 1826, 19.23 2208, a1bi 363). (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 2422	An equivalence related to implicit substitution. Usage of this theorem is discouraged because it depends on ax-13 2374. See equsalhw 2292 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 2423	An equivalence related to implicit substitution. Usage of this theorem is discouraged because it depends on ax-13 2374. See equsexhv 2293 for a version with a disjoint variable condition which does not require ax-13 2374. (Contributed by NM, 5-Aug-1993.) (New usage is discouraged.)
⊢ (𝜓 → ∀𝑥𝜓)    &   ⊢ (𝑥 = 𝑦 → (𝜑 ↔ 𝜓))    ⇒   ⊢ (∃𝑥(𝑥 = 𝑦 ∧ 𝜑) ↔ 𝜓)
Theorem	axc15 2424	Derivation of set.mm's original ax-c15 36897 from ax-c11n 36896 and the shorter ax-12 2175 that has replaced it. Theorem ax12 2425 shows the reverse derivation of ax-12 2175 from ax-c15 36897. Normally, axc15 2424 should be used rather than ax-c15 36897, except by theorems specifically studying the latter's properties. Usage of this theorem is discouraged because it depends on ax-13 2374. (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 2425	Rederivation of Axiom ax-12 2175 from ax12v 2176 (used only via sp 2180) , axc11r 2368, and axc15 2424 (on top of Tarski's FOL). Since this version depends on ax-13 2374, usage of the weaker ax12v 2176, ax12w 2133, ax12i 1974 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 2426	A bidirectional version of axc15 2424. Usage of this theorem is discouraged because it depends on ax-13 2374. (Contributed by NM, 30-Jun-2006.) (New usage is discouraged.)
⊢ ((¬ ∀𝑥 𝑥 = 𝑦 ∧ 𝑥 = 𝑦) → (𝜑 ↔ ∀𝑥(𝑥 = 𝑦 → 𝜑)))
Theorem	ax13ALT 2427	Alternate proof of ax13 2377 from FOL, sp 2180, and axc9 2384. (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 2428	Derive set.mm's original ax-c11n 36896 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 2062. Use aecom 2429 instead when this does not lengthen the proof. Usage of this theorem is discouraged because it depends on ax-13 2374. (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 2429	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 2374. (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 2430	A commutation rule for identical variable specifiers. Usage of this theorem is discouraged because it depends on ax-13 2374. (Contributed by NM, 10-May-1993.) (New usage is discouraged.)
⊢ (∀𝑥 𝑥 = 𝑦 → 𝜑)    ⇒   ⊢ (∀𝑦 𝑦 = 𝑥 → 𝜑)
Theorem	naecoms 2431	A commutation rule for distinct variable specifiers. Usage of this theorem is discouraged because it depends on ax-13 2374. (Contributed by NM, 2-Jan-2002.) (New usage is discouraged.)
⊢ (¬ ∀𝑥 𝑥 = 𝑦 → 𝜑)    ⇒   ⊢ (¬ ∀𝑦 𝑦 = 𝑥 → 𝜑)
Theorem	axc11 2432	Show that ax-c11 36895 can be derived from ax-c11n 36896 in the form of axc11n 2428. Normally, axc11 2432 should be used rather than ax-c11 36895, except by theorems specifically studying the latter's properties. Usage of this theorem is discouraged because it depends on ax-13 2374. 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))
Theorem	hbae 2433	All variables are effectively bound in an identical variable specifier. Usage of this theorem is discouraged because it depends on ax-13 2374. Use the weaker hbaev 2066 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 2434	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 2374. Use the weaker hbnaev 2069 when possible. (Contributed by NM, 13-May-1993.) (New usage is discouraged.)
⊢ (¬ ∀𝑥 𝑥 = 𝑦 → ∀𝑧 ¬ ∀𝑥 𝑥 = 𝑦)
Theorem	nfae 2435	All variables are effectively bound in an identical variable specifier. Usage of this theorem is discouraged because it depends on ax-13 2374. (Contributed by Mario Carneiro, 11-Aug-2016.) (New usage is discouraged.)
⊢ Ⅎ𝑧∀𝑥 𝑥 = 𝑦
Theorem	nfnae 2436	All variables are effectively bound in a distinct variable specifier. Usage of this theorem is discouraged because it depends on ax-13 2374. Use the weaker nfnaew 2149 when possible. (Contributed by Mario Carneiro, 11-Aug-2016.) (New usage is discouraged.)
⊢ Ⅎ𝑧 ¬ ∀𝑥 𝑥 = 𝑦
Theorem	hbnaes 2437	Rule that applies hbnae 2434 to antecedent. Usage of this theorem is discouraged because it depends on ax-13 2374. (Contributed by NM, 15-May-1993.) (New usage is discouraged.)
⊢ (∀𝑧 ¬ ∀𝑥 𝑥 = 𝑦 → 𝜑)    ⇒   ⊢ (¬ ∀𝑥 𝑥 = 𝑦 → 𝜑)
Theorem	axc16i 2438*	Inference with axc16 2257 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 2374. Use axc16 2257 instead. (New usage is discouraged.)
⊢ (𝑥 = 𝑧 → (𝜑 ↔ 𝜓))    &   ⊢ (𝜓 → ∀𝑥𝜓)    ⇒   ⊢ (∀𝑥 𝑥 = 𝑦 → (𝜑 → ∀𝑥𝜑))
Theorem	axc16nfALT 2439*	Alternate proof of axc16nf 2259, shorter but requiring ax-11 2158 and ax-13 2374. (Contributed by Mario Carneiro, 7-Oct-2016.) (Proof modification is discouraged.) (New usage is discouraged.)
⊢ (∀𝑥 𝑥 = 𝑦 → Ⅎ𝑧𝜑)
Theorem	dral2 2440	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 2374. Usage of albidv 1927 is preferred, which requires fewer axioms. (Contributed by NM, 27-Feb-2005.) Allow a shortening of dral1 2441. (Revised by Wolf Lammen, 4-Mar-2018.) (New usage is discouraged.)
⊢ (∀𝑥 𝑥 = 𝑦 → (𝜑 ↔ 𝜓))    ⇒   ⊢ (∀𝑥 𝑥 = 𝑦 → (∀𝑧𝜑 ↔ ∀𝑧𝜓))
Theorem	dral1 2441	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 2374. Use the weaker dral1v 2369 if possible. (Contributed by NM, 24-Nov-1994.) Remove dependency on ax-11 2158. (Revised by Wolf Lammen, 6-Sep-2018.) (New usage is discouraged.)
⊢ (∀𝑥 𝑥 = 𝑦 → (𝜑 ↔ 𝜓))    ⇒   ⊢ (∀𝑥 𝑥 = 𝑦 → (∀𝑥𝜑 ↔ ∀𝑦𝜓))
Theorem	dral1ALT 2442	Alternate proof of dral1 2441, shorter but requiring ax-11 2158. (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 2443	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 2374. Use the weaker drex1v 2371 if possible. (Contributed by NM, 27-Feb-2005.) (New usage is discouraged.)
⊢ (∀𝑥 𝑥 = 𝑦 → (𝜑 ↔ 𝜓))    ⇒   ⊢ (∀𝑥 𝑥 = 𝑦 → (∃𝑥𝜑 ↔ ∃𝑦𝜓))
Theorem	drex2 2444	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 2374. Usage of exbidv 1928 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 2445	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 2374. Use drnf1v 2372 instead. (New usage is discouraged.)
⊢ (∀𝑥 𝑥 = 𝑦 → (𝜑 ↔ 𝜓))    ⇒   ⊢ (∀𝑥 𝑥 = 𝑦 → (Ⅎ𝑥𝜑 ↔ Ⅎ𝑦𝜓))
Theorem	drnf2 2446	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 2374. Use nfbidv 1929 instead. (New usage is discouraged.)
⊢ (∀𝑥 𝑥 = 𝑦 → (𝜑 ↔ 𝜓))    ⇒   ⊢ (∀𝑥 𝑥 = 𝑦 → (Ⅎ𝑧𝜑 ↔ Ⅎ𝑧𝜓))
Theorem	nfald2 2447	Variation on nfald 2326 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 2374. Use nfald 2326 instead. (New usage is discouraged.)
⊢ Ⅎ𝑦𝜑    &   ⊢ ((𝜑 ∧ ¬ ∀𝑥 𝑥 = 𝑦) → Ⅎ𝑥𝜓)    ⇒   ⊢ (𝜑 → Ⅎ𝑥∀𝑦𝜓)
Theorem	nfexd2 2448	Variation on nfexd 2327 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 2374. Use nfexd 2327 instead. (New usage is discouraged.)
⊢ Ⅎ𝑦𝜑    &   ⊢ ((𝜑 ∧ ¬ ∀𝑥 𝑥 = 𝑦) → Ⅎ𝑥𝜓)    ⇒   ⊢ (𝜑 → Ⅎ𝑥∃𝑦𝜓)
Theorem	exdistrf 2449	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 2374. Use exdistr 1962 instead. (New usage is discouraged.)
⊢ (¬ ∀𝑥 𝑥 = 𝑦 → Ⅎ𝑦𝜑)    ⇒   ⊢ (∃𝑥∃𝑦(𝜑 ∧ 𝜓) → ∃𝑥(𝜑 ∧ ∃𝑦𝜓))
Theorem	dvelimf 2450	Version of dvelimv 2454 without any variable restrictions. Usage of this theorem is discouraged because it depends on ax-13 2374. (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 2451	Deduction form of dvelimf 2450. Usage of this theorem is discouraged because it depends on ax-13 2374. (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 2452	Version of dvelim 2453 without any variable restrictions. Usage of this theorem is discouraged because it depends on ax-13 2374. Check out dvelimhw 2347 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 2453*	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 2451. Other variants of this theorem are dvelimh 2452 (with no distinct variable restrictions) and dvelimhw 2347 (that avoids ax-13 2374). Usage of this theorem is discouraged because it depends on ax-13 2374. Check out dvelimhw 2347 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 2454*	Similar to dvelim 2453 with first hypothesis replaced by a distinct variable condition. Usage of this theorem is discouraged because it depends on ax-13 2374. Check out dvelimhw 2347 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 2455*	Version of dvelim 2453 using "not free" notation. Usage of this theorem is discouraged because it depends on ax-13 2374. (Contributed by Mario Carneiro, 9-Oct-2016.) (New usage is discouraged.)
⊢ Ⅎ𝑥𝜑    &   ⊢ (𝑧 = 𝑦 → (𝜑 ↔ 𝜓))    ⇒   ⊢ (¬ ∀𝑥 𝑥 = 𝑦 → Ⅎ𝑥𝜓)
Theorem	dveeq2ALT 2456*	Alternate proof of dveeq2 2380, shorter but requiring ax-11 2158. (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 2457	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 2374. See equvinv 2036 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 2458	A variable elimination law for equality with no distinct variable requirements. Compare equvini 2457. Usage of this theorem is discouraged because it depends on ax-13 2374. Use equvelv 2038 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 2459	A property related to substitution that unlike equs5 2462 does not require a distinctor antecedent. Usage of this theorem is discouraged because it depends on ax-13 2374. This proof uses ax12 2425, see equs5aALT 2366 for an alternative one using ax-12 2175 but not ax13 2377. Usage of the weaker equs5av 2275 is preferred, which uses ax12v2 2177, but not ax-13 2374. (Contributed by NM, 2-Feb-2007.) (New usage is discouraged.)
⊢ (∃𝑥(𝑥 = 𝑦 ∧ ∀𝑦𝜑) → ∀𝑥(𝑥 = 𝑦 → 𝜑))
Theorem	equs5e 2460	A property related to substitution that unlike equs5 2462 does not require a distinctor antecedent. This proof uses ax12 2425, see equs5eALT 2367 for an alternative one using ax-12 2175 but not ax13 2377. Usage of this theorem is discouraged because it depends on ax-13 2374. (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 2461	Two ways of expressing substitution when 𝑦 is not free in 𝜑. The implication "to the left" is equs4 2418 and does not require the nonfreeness hypothesis. Theorem sbalex 2239 replaces the nonfreeness hypothesis with a disjoint variable condition and equs5 2462 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 2374. Use sbalex 2239 instead. (New usage is discouraged.)
⊢ Ⅎ𝑦𝜑    ⇒   ⊢ (∃𝑥(𝑥 = 𝑦 ∧ 𝜑) ↔ ∀𝑥(𝑥 = 𝑦 → 𝜑))
Theorem	equs5 2462	Lemma used in proofs of substitution properties. If there is a disjoint variable condition on 𝑥, 𝑦, then sbalex 2239 can be used instead; if 𝑦 is not free in 𝜑, then equs45f 2461 can be used. Usage of this theorem is discouraged because it depends on ax-13 2374. (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 2463*	Quantifier introduction when one pair of variables is disjoint. Usage of this theorem is discouraged because it depends on ax-13 2374. (Contributed by NM, 2-Jan-2002.) (New usage is discouraged.)
⊢ (¬ ∀𝑥 𝑥 = 𝑦 → (𝑦 ∈ 𝑧 → ∀𝑥 𝑦 ∈ 𝑧))
Theorem	dveel2 2464*	Quantifier introduction when one pair of variables is disjoint. Usage of this theorem is discouraged because it depends on ax-13 2374. (Contributed by NM, 2-Jan-2002.) (New usage is discouraged.)
⊢ (¬ ∀𝑥 𝑥 = 𝑦 → (𝑧 ∈ 𝑦 → ∀𝑥 𝑧 ∈ 𝑦))
Theorem	axc14 2465	Axiom ax-c14 36899 is redundant if we assume ax-5 1917. 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 2464 and ax-5 1917. 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 2374. (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 2466	Equivalence involving substitution for a variable not free. Usage of this theorem is discouraged because it depends on ax-13 2374. Usage of sb6 2092 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 2467	Substitution does not change an identical variable specifier. Usage of this theorem is discouraged because it depends on ax-13 2374. (Contributed by NM, 15-May-1993.) (New usage is discouraged.)
⊢ ([𝑤 / 𝑧]∀𝑥 𝑥 = 𝑦 ↔ ∀𝑥 𝑥 = 𝑦)
Theorem	sbequ6 2468	Substitution does not change a distinctor. Usage of this theorem is discouraged because it depends on ax-13 2374. (Contributed by NM, 5-Aug-1993.) (New usage is discouraged.)
⊢ ([𝑤 / 𝑧] ¬ ∀𝑥 𝑥 = 𝑦 ↔ ¬ ∀𝑥 𝑥 = 𝑦)
Theorem	sb5rf 2469	Reversed substitution. Usage of this theorem is discouraged because it depends on ax-13 2374. (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.)
⊢ Ⅎ𝑦𝜑    ⇒   ⊢ (𝜑 ↔ ∃𝑦(𝑦 = 𝑥 ∧ [𝑦 / 𝑥]𝜑))
Theorem	sb6rf 2470	Reversed substitution. For a version requiring disjoint variables, but fewer axioms, see sb6rfv 2357. Usage of this theorem is discouraged because it depends on ax-13 2374. Use the weaker sb6rfv 2357 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.)
⊢ Ⅎ𝑦𝜑    ⇒   ⊢ (𝜑 ↔ ∀𝑦(𝑦 = 𝑥 → [𝑦 / 𝑥]𝜑))
Theorem	ax12vALT 2471*	Alternate proof of ax12v2 2177, 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 = 𝑦 → 𝜑)))
Theorem	2ax6elem 2472	We can always find values matching 𝑥 and 𝑦, as long as they are represented by distinct variables. This theorem merges two ax6e 2385 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 42146. Usage of this theorem is discouraged because it depends on ax-13 2374. (Contributed by Wolf Lammen, 27-Sep-2018.) (New usage is discouraged.)
⊢ (¬ ∀𝑤 𝑤 = 𝑧 → ∃𝑧∃𝑤(𝑧 = 𝑥 ∧ 𝑤 = 𝑦))
Theorem	2ax6e 2473*	We can always find values matching 𝑥 and 𝑦, as long as they are represented by distinct variables. Version of 2ax6elem 2472 with a distinct variable constraint. Usage of this theorem is discouraged because it depends on ax-13 2374. (Contributed by Wolf Lammen, 28-Sep-2018.) (Proof shortened by Wolf Lammen, 3-Oct-2023.) (New usage is discouraged.)
⊢ ∃𝑧∃𝑤(𝑧 = 𝑥 ∧ 𝑤 = 𝑦)
Theorem	2sb5rf 2474*	Reversed double substitution. Usage of this theorem is discouraged because it depends on ax-13 2374. (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.)
⊢ Ⅎ𝑧𝜑    &   ⊢ Ⅎ𝑤𝜑    ⇒   ⊢ (𝜑 ↔ ∃𝑧∃𝑤((𝑧 = 𝑥 ∧ 𝑤 = 𝑦) ∧ [𝑧 / 𝑥][𝑤 / 𝑦]𝜑))
Theorem	2sb6rf 2475*	Reversed double substitution. Usage of this theorem is discouraged because it depends on ax-13 2374. (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.)
⊢ Ⅎ𝑧𝜑    &   ⊢ Ⅎ𝑤𝜑    ⇒   ⊢ (𝜑 ↔ ∀𝑧∀𝑤((𝑧 = 𝑥 ∧ 𝑤 = 𝑦) → [𝑧 / 𝑥][𝑤 / 𝑦]𝜑))
Theorem	sbel2x 2476*	Elimination of double substitution. Usage of this theorem is discouraged because it depends on ax-13 2374. (Contributed by NM, 5-Aug-1993.) (Proof shortened by Wolf Lammen, 29-Sep-2018.) (New usage is discouraged.)
⊢ (𝜑 ↔ ∃𝑥∃𝑦((𝑥 = 𝑧 ∧ 𝑦 = 𝑤) ∧ [𝑦 / 𝑤][𝑥 / 𝑧]𝜑))
Theorem	sb4b 2477	Simplified definition of substitution when variables are distinct. Version of sb6 2092 with a distinctor antecedent. Usage of this theorem is discouraged because it depends on ax-13 2374. (Contributed by NM, 27-May-1997.) Revise df-sb 2072. (Revised by Wolf Lammen, 21-Feb-2024.) (New usage is discouraged.)
⊢ (¬ ∀𝑥 𝑥 = 𝑡 → ([𝑡 / 𝑥]𝜑 ↔ ∀𝑥(𝑥 = 𝑡 → 𝜑)))
Theorem	sb4bOLD 2478	Obsolete version of sb4b 2477 as of 21-Feb-2024. (Contributed by NM, 27-May-1997.) Revise df-sb 2072. (Revised by Wolf Lammen, 25-Jul-2023.) (New usage is discouraged.) (Proof modification is discouraged.)
⊢ (¬ ∀𝑥 𝑥 = 𝑡 → ([𝑡 / 𝑥]𝜑 ↔ ∀𝑥(𝑥 = 𝑡 → 𝜑)))
Theorem	sb3b 2479	Simplified definition of substitution when variables are distinct. This is the biconditional strengthening of sb3 2480. Usage of this theorem is discouraged because it depends on ax-13 2374. (Contributed by BJ, 6-Oct-2018.) Shorten sb3 2480. (Revised by Wolf Lammen, 21-Feb-2021.) (New usage is discouraged.)
⊢ (¬ ∀𝑥 𝑥 = 𝑦 → ([𝑦 / 𝑥]𝜑 ↔ ∃𝑥(𝑥 = 𝑦 ∧ 𝜑)))
Theorem	sb3 2480	One direction of a simplified definition of substitution when variables are distinct. Usage of this theorem is discouraged because it depends on ax-13 2374. (Contributed by NM, 5-Aug-1993.) (Proof shortened by Wolf Lammen, 21-Feb-2024.) (New usage is discouraged.)
⊢ (¬ ∀𝑥 𝑥 = 𝑦 → (∃𝑥(𝑥 = 𝑦 ∧ 𝜑) → [𝑦 / 𝑥]𝜑))
Theorem	sb1 2481	One direction of a simplified definition of substitution. The converse requires either a disjoint variable condition (sb5 2272) or a nonfreeness hypothesis (sb5f 2504). Usage of this theorem is discouraged because it depends on ax-13 2374. Use the weaker sb1v 2094 when possible. (Contributed by NM, 13-May-1993.) Revise df-sb 2072. (Revised by Wolf Lammen, 21-Feb-2024.) (New usage is discouraged.)
⊢ ([𝑦 / 𝑥]𝜑 → ∃𝑥(𝑥 = 𝑦 ∧ 𝜑))
Theorem	sb2 2482	One direction of a simplified definition of substitution. The converse requires either a disjoint variable condition (sb6 2092) or a nonfreeness hypothesis (sb6f 2503). Usage of this theorem is discouraged because it depends on ax-13 2374. (Contributed by NM, 13-May-1993.) Revise df-sb 2072. (Revised by Wolf Lammen, 26-Jul-2023.) (New usage is discouraged.)
⊢ (∀𝑥(𝑥 = 𝑦 → 𝜑) → [𝑦 / 𝑥]𝜑)
Theorem	sb3OLD 2483	Obsolete version of sb3 2480 as of 21-Feb-2024. (Contributed by NM, 5-Aug-1993.) (Proof modification is discouraged.) (New usage is discouraged.)
⊢ (¬ ∀𝑥 𝑥 = 𝑦 → (∃𝑥(𝑥 = 𝑦 ∧ 𝜑) → [𝑦 / 𝑥]𝜑))
Theorem	sb1OLD 2484	Obsolete version of sb1 2481 as of 21-Feb-2024. (Contributed by NM, 13-May-1993.) Revise df-sb 2072. (Revised by Wolf Lammen, 29-Jul-2023.) (Proof modification is discouraged.) (New usage is discouraged.)
⊢ ([𝑦 / 𝑥]𝜑 → ∃𝑥(𝑥 = 𝑦 ∧ 𝜑))
Theorem	sb3bOLD 2485	Obsolete version of sb3b 2479 as of 21-Feb-2024. (Contributed by BJ, 6-Oct-2018.) (Proof modification is discouraged.) (New usage is discouraged.)
⊢ (¬ ∀𝑥 𝑥 = 𝑦 → ([𝑦 / 𝑥]𝜑 ↔ ∃𝑥(𝑥 = 𝑦 ∧ 𝜑)))
Theorem	sb4a 2486	A version of one implication of sb4b 2477 that does not require a distinctor antecedent. Usage of this theorem is discouraged because it depends on ax-13 2374. Use the weaker sb4av 2240 when possible. (Contributed by NM, 2-Feb-2007.) Revise df-sb 2072. (Revised by Wolf Lammen, 28-Jul-2023.) (New usage is discouraged.)
⊢ ([𝑡 / 𝑥]∀𝑡𝜑 → ∀𝑥(𝑥 = 𝑡 → 𝜑))
Theorem	dfsb1 2487	Alternate definition of substitution. Remark 9.1 in [Megill] p. 447 (p. 15 of the preprint). This was the original definition before df-sb 2072. Note that it does not require dummy variables in its definiens; this is done by having 𝑥 free in the first conjunct and bound in the second. Usage of this theorem is discouraged because it depends on ax-13 2374. (Contributed by BJ, 9-Jul-2023.) Revise df-sb 2072. (Revised by Wolf Lammen, 29-Jul-2023.) (New usage is discouraged.)
⊢ ([𝑦 / 𝑥]𝜑 ↔ ((𝑥 = 𝑦 → 𝜑) ∧ ∃𝑥(𝑥 = 𝑦 ∧ 𝜑)))
Theorem	hbsb2 2488	Bound-variable hypothesis builder for substitution. Usage of this theorem is discouraged because it depends on ax-13 2374. (Contributed by NM, 14-May-1993.) (New usage is discouraged.)
⊢ (¬ ∀𝑥 𝑥 = 𝑦 → ([𝑦 / 𝑥]𝜑 → ∀𝑥[𝑦 / 𝑥]𝜑))
Theorem	nfsb2 2489	Bound-variable hypothesis builder for substitution. Usage of this theorem is discouraged because it depends on ax-13 2374. (Contributed by Mario Carneiro, 4-Oct-2016.) (New usage is discouraged.)
⊢ (¬ ∀𝑥 𝑥 = 𝑦 → Ⅎ𝑥[𝑦 / 𝑥]𝜑)
Theorem	hbsb2a 2490	Special case of a bound-variable hypothesis builder for substitution. Usage of this theorem is discouraged because it depends on ax-13 2374. (Contributed by NM, 2-Feb-2007.) (New usage is discouraged.)
⊢ ([𝑦 / 𝑥]∀𝑦𝜑 → ∀𝑥[𝑦 / 𝑥]𝜑)
Theorem	sb4e 2491	One direction of a simplified definition of substitution that unlike sb4b 2477 does not require a distinctor antecedent. Usage of this theorem is discouraged because it depends on ax-13 2374. (Contributed by NM, 2-Feb-2007.) (New usage is discouraged.)
⊢ ([𝑦 / 𝑥]𝜑 → ∀𝑥(𝑥 = 𝑦 → ∃𝑦𝜑))
Theorem	hbsb2e 2492	Special case of a bound-variable hypothesis builder for substitution. Usage of this theorem is discouraged because it depends on ax-13 2374. (Contributed by NM, 2-Feb-2007.) (New usage is discouraged.)
⊢ ([𝑦 / 𝑥]𝜑 → ∀𝑥[𝑦 / 𝑥]∃𝑦𝜑)
Theorem	hbsb3 2493	If 𝑦 is not free in 𝜑, 𝑥 is not free in [𝑦 / 𝑥]𝜑. Usage of this theorem is discouraged because it depends on ax-13 2374. Check out bj-hbsb3v 34991 for a weaker version requiring fewer axioms. (Contributed by NM, 14-May-1993.) (New usage is discouraged.)
⊢ (𝜑 → ∀𝑦𝜑)    ⇒   ⊢ ([𝑦 / 𝑥]𝜑 → ∀𝑥[𝑦 / 𝑥]𝜑)
Theorem	nfs1 2494	If 𝑦 is not free in 𝜑, 𝑥 is not free in [𝑦 / 𝑥]𝜑. Usage of this theorem is discouraged because it depends on ax-13 2374. Check out nfs1v 2157 for a version requiring fewer axioms. (Contributed by Mario Carneiro, 11-Aug-2016.) (New usage is discouraged.)
⊢ Ⅎ𝑦𝜑    ⇒   ⊢ Ⅎ𝑥[𝑦 / 𝑥]𝜑
Theorem	axc16ALT 2495*	Alternate proof of axc16 2257, shorter but requiring ax-10 2141, ax-11 2158, ax-13 2374 and using df-nf 1791 and df-sb 2072. (Contributed by NM, 17-May-2008.) (Proof modification is discouraged.) (New usage is discouraged.)
⊢ (∀𝑥 𝑥 = 𝑦 → (𝜑 → ∀𝑥𝜑))
Theorem	axc16gALT 2496*	Alternate proof of axc16g 2256 that uses df-sb 2072 and requires ax-10 2141, ax-11 2158, ax-13 2374. (Contributed by NM, 15-May-1993.) (Proof shortened by Andrew Salmon, 25-May-2011.) (Proof modification is discouraged.) (New usage is discouraged.)
⊢ (∀𝑥 𝑥 = 𝑦 → (𝜑 → ∀𝑧𝜑))
Theorem	equsb1 2497	Substitution applied to an atomic wff. Usage of this theorem is discouraged because it depends on ax-13 2374. Use the weaker equsb1v 2107 if possible. (Contributed by NM, 10-May-1993.) (New usage is discouraged.)
⊢ [𝑦 / 𝑥]𝑥 = 𝑦
Theorem	equsb2 2498	Substitution applied to an atomic wff. Usage of this theorem is discouraged because it depends on ax-13 2374. Check out equsb1v 2107 for a version requiring fewer axioms. (Contributed by NM, 10-May-1993.) (New usage is discouraged.)
⊢ [𝑦 / 𝑥]𝑦 = 𝑥
Theorem	dfsb2 2499	An alternate definition of proper substitution that, like dfsb1 2487, mixes free and bound variables to avoid distinct variable requirements. Usage of this theorem is discouraged because it depends on ax-13 2374. (Contributed by NM, 17-Feb-2005.) (New usage is discouraged.)
⊢ ([𝑦 / 𝑥]𝜑 ↔ ((𝑥 = 𝑦 ∧ 𝜑) ∨ ∀𝑥(𝑥 = 𝑦 → 𝜑)))
Theorem	dfsb3 2500	An alternate definition of proper substitution df-sb 2072 that uses only primitive connectives (no defined terms) on the right-hand side. Usage of this theorem is discouraged because it depends on ax-13 2374. (Contributed by NM, 6-Mar-2007.) (New usage is discouraged.)
⊢ ([𝑦 / 𝑥]𝜑 ↔ ((𝑥 = 𝑦 → ¬ 𝜑) → ∀𝑥(𝑥 = 𝑦 → 𝜑)))