P. 23 of Theorem List - Metamath Proof Explorer

Theorem List for Metamath Proof Explorer - 2201-2300   *Has distinct variable group(s)

Type	Label	Description
Statement
Theorem	qexmid 2201	Quantified excluded middle (see exmid 895). Also known as the drinker paradox (if 𝜑(𝑥) is interpreted as "𝑥 drinks", then this theorem tells that there exists a person such that, if this person drinks, then everyone drinks). Exercise 9.2a of Boolos, p. 111, Computability and Logic. (Contributed by NM, 10-Dec-2000.)
⊢ ∃𝑥(𝜑 → ∀𝑥𝜑)
Theorem	nf5r 2202	Consequence of the definition of not-free. (Contributed by Mario Carneiro, 26-Sep-2016.) df-nf 1786 changed. (Revised by Wolf Lammen, 11-Sep-2021.) (Proof shortened by Wolf Lammen, 23-Nov-2023.)
⊢ (Ⅎ𝑥𝜑 → (𝜑 → ∀𝑥𝜑))
Theorem	nf5ri 2203	Consequence of the definition of not-free. (Contributed by Mario Carneiro, 11-Aug-2016.) (Proof shortened by Wolf Lammen, 15-Mar-2023.)
⊢ Ⅎ𝑥𝜑    ⇒   ⊢ (𝜑 → ∀𝑥𝜑)
Theorem	nf5rd 2204	Consequence of the definition of not-free in a context. (Contributed by Mario Carneiro, 11-Aug-2016.)
⊢ (𝜑 → Ⅎ𝑥𝜓)    ⇒   ⊢ (𝜑 → (𝜓 → ∀𝑥𝜓))
Theorem	spimedv 2205*	Deduction version of spimev 2397. Version of spimed 2393 with a disjoint variable condition, which does not require ax-13 2377. See spime 2394 for a non-deduction version. (Contributed by NM, 14-May-1993.) (Revised by BJ, 31-May-2019.)
⊢ (𝜒 → Ⅎ𝑥𝜑)    &   ⊢ (𝑥 = 𝑦 → (𝜑 → 𝜓))    ⇒   ⊢ (𝜒 → (𝜑 → ∃𝑥𝜓))
Theorem	spimefv 2206*	Version of spime 2394 with a disjoint variable condition, which does not require ax-13 2377. (Contributed by BJ, 31-May-2019.)
⊢ Ⅎ𝑥𝜑    &   ⊢ (𝑥 = 𝑦 → (𝜑 → 𝜓))    ⇒   ⊢ (𝜑 → ∃𝑥𝜓)
Theorem	nfim1 2207	A closed form of nfim 1898. (Contributed by NM, 2-Jun-1993.) (Revised by Mario Carneiro, 24-Sep-2016.) (Proof shortened by Wolf Lammen, 2-Jan-2018.) df-nf 1786 changed. (Revised by Wolf Lammen, 18-Sep-2021.)
⊢ Ⅎ𝑥𝜑    &   ⊢ (𝜑 → Ⅎ𝑥𝜓)    ⇒   ⊢ Ⅎ𝑥(𝜑 → 𝜓)
Theorem	nfan1 2208	A closed form of nfan 1901. (Contributed by Mario Carneiro, 3-Oct-2016.) df-nf 1786 changed. (Revised by Wolf Lammen, 18-Sep-2021.) (Proof shortened by Wolf Lammen, 7-Jul-2022.)
⊢ Ⅎ𝑥𝜑    &   ⊢ (𝜑 → Ⅎ𝑥𝜓)    ⇒   ⊢ Ⅎ𝑥(𝜑 ∧ 𝜓)
Theorem	19.3t 2209	Closed form of 19.3 2210 and version of 19.9t 2212 with a universal quantifier. (Contributed by NM, 9-Nov-2020.) (Proof shortened by BJ, 9-Oct-2022.)
⊢ (Ⅎ𝑥𝜑 → (∀𝑥𝜑 ↔ 𝜑))
Theorem	19.3 2210	A wff may be quantified with a variable not free in it. Version of 19.9 2213 with a universal quantifier. Theorem 19.3 of [Margaris] p. 89. See 19.3v 1984 for a version requiring fewer axioms. (Contributed by NM, 12-Mar-1993.) (Revised by Mario Carneiro, 24-Sep-2016.)
⊢ Ⅎ𝑥𝜑    ⇒   ⊢ (∀𝑥𝜑 ↔ 𝜑)
Theorem	19.9d 2211	A deduction version of one direction of 19.9 2213. (Contributed by NM, 14-May-1993.) (Revised by Mario Carneiro, 24-Sep-2016.) Revised to shorten other proofs. (Revised by Wolf Lammen, 14-Jul-2020.) df-nf 1786 changed. (Revised by Wolf Lammen, 11-Sep-2021.) (Proof shortened by Wolf Lammen, 8-Jul-2022.)
⊢ (𝜓 → Ⅎ𝑥𝜑)    ⇒   ⊢ (𝜓 → (∃𝑥𝜑 → 𝜑))
Theorem	19.9t 2212	Closed form of 19.9 2213 and version of 19.3t 2209 with an existential quantifier. (Contributed by NM, 13-May-1993.) (Revised by Mario Carneiro, 24-Sep-2016.) (Proof shortened by Wolf Lammen, 14-Jul-2020.)
⊢ (Ⅎ𝑥𝜑 → (∃𝑥𝜑 ↔ 𝜑))
Theorem	19.9 2213	A wff may be existentially quantified with a variable not free in it. Version of 19.3 2210 with an existential quantifier. Theorem 19.9 of [Margaris] p. 89. See 19.9v 1986 for a version requiring fewer axioms. (Contributed by FL, 24-Mar-2007.) (Revised by Mario Carneiro, 24-Sep-2016.) (Proof shortened by Wolf Lammen, 30-Dec-2017.) Revised to shorten other proofs. (Revised by Wolf Lammen, 14-Jul-2020.)
⊢ Ⅎ𝑥𝜑    ⇒   ⊢ (∃𝑥𝜑 ↔ 𝜑)
Theorem	19.21t 2214	Closed form of Theorem 19.21 of [Margaris] p. 90, see 19.21 2215. (Contributed by NM, 27-May-1997.) (Revised by Mario Carneiro, 24-Sep-2016.) (Proof shortened by Wolf Lammen, 3-Jan-2018.) df-nf 1786 changed. (Revised by Wolf Lammen, 11-Sep-2021.) (Proof shortened by BJ, 3-Nov-2021.)
⊢ (Ⅎ𝑥𝜑 → (∀𝑥(𝜑 → 𝜓) ↔ (𝜑 → ∀𝑥𝜓)))
Theorem	19.21 2215	Theorem 19.21 of [Margaris] p. 90. The hypothesis can be thought of as "𝑥 is not free in 𝜑". See 19.21v 1941 for a version requiring fewer axioms. See also 19.21h 2294. (Contributed by NM, 14-May-1993.) (Revised by Mario Carneiro, 24-Sep-2016.) df-nf 1786 changed. (Revised by Wolf Lammen, 18-Sep-2021.)
⊢ Ⅎ𝑥𝜑    ⇒   ⊢ (∀𝑥(𝜑 → 𝜓) ↔ (𝜑 → ∀𝑥𝜓))
Theorem	stdpc5 2216	An axiom scheme of standard predicate calculus that emulates Axiom 5 of [Mendelson] p. 69. The hypothesis Ⅎ𝑥𝜑 can be thought of as emulating "𝑥 is not free in 𝜑". With this definition, the meaning of "not free" is less restrictive than the usual textbook definition; for example 𝑥 would not (for us) be free in 𝑥 = 𝑥 by nfequid 2015. This theorem scheme can be proved as a metatheorem of Mendelson's axiom system, even though it is slightly stronger than his Axiom 5. See stdpc5v 1940 for a version requiring fewer axioms. (Contributed by NM, 22-Sep-1993.) (Revised by Mario Carneiro, 12-Oct-2016.) (Proof shortened by Wolf Lammen, 1-Jan-2018.) Remove dependency on ax-10 2147. (Revised by Wolf Lammen, 4-Jul-2021.) (Proof shortened by Wolf Lammen, 11-Oct-2021.)
⊢ Ⅎ𝑥𝜑    ⇒   ⊢ (∀𝑥(𝜑 → 𝜓) → (𝜑 → ∀𝑥𝜓))
Theorem	19.21-2 2217	Version of 19.21 2215 with two quantifiers. (Contributed by NM, 4-Feb-2005.)
⊢ Ⅎ𝑥𝜑    &   ⊢ Ⅎ𝑦𝜑    ⇒   ⊢ (∀𝑥∀𝑦(𝜑 → 𝜓) ↔ (𝜑 → ∀𝑥∀𝑦𝜓))
Theorem	19.23t 2218	Closed form of Theorem 19.23 of [Margaris] p. 90. See 19.23 2219. (Contributed by NM, 7-Nov-2005.) (Proof shortened by Wolf Lammen, 13-Aug-2020.) df-nf 1786 changed. (Revised by Wolf Lammen, 11-Sep-2021.) (Proof shortened by BJ, 8-Oct-2022.)
⊢ (Ⅎ𝑥𝜓 → (∀𝑥(𝜑 → 𝜓) ↔ (∃𝑥𝜑 → 𝜓)))
Theorem	19.23 2219	Theorem 19.23 of [Margaris] p. 90. See 19.23v 1944 for a version requiring fewer axioms. (Contributed by NM, 24-Jan-1993.) (Revised by Mario Carneiro, 24-Sep-2016.)
⊢ Ⅎ𝑥𝜓    ⇒   ⊢ (∀𝑥(𝜑 → 𝜓) ↔ (∃𝑥𝜑 → 𝜓))
Theorem	alimd 2220	Deduction form of Theorem 19.20 of [Margaris] p. 90, see alim 1812. See alimdh 1819, alimdv 1918 for variants requiring fewer axioms. (Contributed by Mario Carneiro, 24-Sep-2016.)
⊢ Ⅎ𝑥𝜑    &   ⊢ (𝜑 → (𝜓 → 𝜒))    ⇒   ⊢ (𝜑 → (∀𝑥𝜓 → ∀𝑥𝜒))
Theorem	alrimi 2221	Inference form of Theorem 19.21 of [Margaris] p. 90, see 19.21 2215. (Contributed by Mario Carneiro, 24-Sep-2016.)
⊢ Ⅎ𝑥𝜑    &   ⊢ (𝜑 → 𝜓)    ⇒   ⊢ (𝜑 → ∀𝑥𝜓)
Theorem	alrimdd 2222	Deduction form of Theorem 19.21 of [Margaris] p. 90, see 19.21 2215. (Contributed by Mario Carneiro, 24-Sep-2016.)
⊢ Ⅎ𝑥𝜑    &   ⊢ (𝜑 → Ⅎ𝑥𝜓)    &   ⊢ (𝜑 → (𝜓 → 𝜒))    ⇒   ⊢ (𝜑 → (𝜓 → ∀𝑥𝜒))
Theorem	alrimd 2223	Deduction form of Theorem 19.21 of [Margaris] p. 90, see 19.21 2215. (Contributed by Mario Carneiro, 24-Sep-2016.)
⊢ Ⅎ𝑥𝜑    &   ⊢ Ⅎ𝑥𝜓    &   ⊢ (𝜑 → (𝜓 → 𝜒))    ⇒   ⊢ (𝜑 → (𝜓 → ∀𝑥𝜒))
Theorem	eximd 2224	Deduction form of Theorem 19.22 of [Margaris] p. 90, see exim 1836. (Contributed by NM, 29-Jun-1993.) (Revised by Mario Carneiro, 24-Sep-2016.)
⊢ Ⅎ𝑥𝜑    &   ⊢ (𝜑 → (𝜓 → 𝜒))    ⇒   ⊢ (𝜑 → (∃𝑥𝜓 → ∃𝑥𝜒))
Theorem	exlimi 2225	Inference associated with 19.23 2219. See exlimiv 1932 for a version with a disjoint variable condition requiring fewer axioms. (Contributed by NM, 10-Jan-1993.) (Revised by Mario Carneiro, 24-Sep-2016.)
⊢ Ⅎ𝑥𝜓    &   ⊢ (𝜑 → 𝜓)    ⇒   ⊢ (∃𝑥𝜑 → 𝜓)
Theorem	exlimd 2226	Deduction form of Theorem 19.9 of [Margaris] p. 89. (Contributed by NM, 23-Jan-1993.) (Revised by Mario Carneiro, 24-Sep-2016.) (Proof shortened by Wolf Lammen, 12-Jan-2018.)
⊢ Ⅎ𝑥𝜑    &   ⊢ Ⅎ𝑥𝜒    &   ⊢ (𝜑 → (𝜓 → 𝜒))    ⇒   ⊢ (𝜑 → (∃𝑥𝜓 → 𝜒))
Theorem	exlimimdd 2227	Existential elimination rule of natural deduction. (Contributed by ML, 17-Jul-2020.) Shorten exlimdd 2228. (Revised by Wolf Lammen, 3-Sep-2023.)
⊢ Ⅎ𝑥𝜑    &   ⊢ Ⅎ𝑥𝜒    &   ⊢ (𝜑 → ∃𝑥𝜓)    &   ⊢ (𝜑 → (𝜓 → 𝜒))    ⇒   ⊢ (𝜑 → 𝜒)
Theorem	exlimdd 2228	Existential elimination rule of natural deduction. (Contributed by Mario Carneiro, 9-Feb-2017.) (Proof shortened by Wolf Lammen, 3-Sep-2023.)
⊢ Ⅎ𝑥𝜑    &   ⊢ Ⅎ𝑥𝜒    &   ⊢ (𝜑 → ∃𝑥𝜓)    &   ⊢ ((𝜑 ∧ 𝜓) → 𝜒)    ⇒   ⊢ (𝜑 → 𝜒)
Theorem	nexd 2229	Deduction for generalization rule for negated wff. (Contributed by Mario Carneiro, 24-Sep-2016.)
⊢ Ⅎ𝑥𝜑    &   ⊢ (𝜑 → ¬ 𝜓)    ⇒   ⊢ (𝜑 → ¬ ∃𝑥𝜓)
Theorem	albid 2230	Formula-building rule for universal quantifier (deduction form). (Contributed by Mario Carneiro, 24-Sep-2016.)
⊢ Ⅎ𝑥𝜑    &   ⊢ (𝜑 → (𝜓 ↔ 𝜒))    ⇒   ⊢ (𝜑 → (∀𝑥𝜓 ↔ ∀𝑥𝜒))
Theorem	exbid 2231	Formula-building rule for existential quantifier (deduction form). (Contributed by Mario Carneiro, 24-Sep-2016.)
⊢ Ⅎ𝑥𝜑    &   ⊢ (𝜑 → (𝜓 ↔ 𝜒))    ⇒   ⊢ (𝜑 → (∃𝑥𝜓 ↔ ∃𝑥𝜒))
Theorem	nfbidf 2232	An equality theorem for effectively not free. (Contributed by Mario Carneiro, 4-Oct-2016.) df-nf 1786 changed. (Revised by Wolf Lammen, 18-Sep-2021.)
⊢ Ⅎ𝑥𝜑    &   ⊢ (𝜑 → (𝜓 ↔ 𝜒))    ⇒   ⊢ (𝜑 → (Ⅎ𝑥𝜓 ↔ Ⅎ𝑥𝜒))
Theorem	19.16 2233	Theorem 19.16 of [Margaris] p. 90. (Contributed by NM, 12-Mar-1993.)
⊢ Ⅎ𝑥𝜑    ⇒   ⊢ (∀𝑥(𝜑 ↔ 𝜓) → (𝜑 ↔ ∀𝑥𝜓))
Theorem	19.17 2234	Theorem 19.17 of [Margaris] p. 90. (Contributed by NM, 12-Mar-1993.)
⊢ Ⅎ𝑥𝜓    ⇒   ⊢ (∀𝑥(𝜑 ↔ 𝜓) → (∀𝑥𝜑 ↔ 𝜓))
Theorem	19.27 2235	Theorem 19.27 of [Margaris] p. 90. See 19.27v 1997 for a version requiring fewer axioms. (Contributed by NM, 21-Jun-1993.)
⊢ Ⅎ𝑥𝜓    ⇒   ⊢ (∀𝑥(𝜑 ∧ 𝜓) ↔ (∀𝑥𝜑 ∧ 𝜓))
Theorem	19.28 2236	Theorem 19.28 of [Margaris] p. 90. See 19.28v 1998 for a version requiring fewer axioms. (Contributed by NM, 1-Aug-1993.) (Proof shortened by Wolf Lammen, 7-May-2025.)
⊢ Ⅎ𝑥𝜑    ⇒   ⊢ (∀𝑥(𝜑 ∧ 𝜓) ↔ (𝜑 ∧ ∀𝑥𝜓))
Theorem	19.19 2237	Theorem 19.19 of [Margaris] p. 90. (Contributed by NM, 12-Mar-1993.)
⊢ Ⅎ𝑥𝜑    ⇒   ⊢ (∀𝑥(𝜑 ↔ 𝜓) → (𝜑 ↔ ∃𝑥𝜓))
Theorem	19.36 2238	Theorem 19.36 of [Margaris] p. 90. See 19.36v 1995 for a version requiring fewer axioms. (Contributed by NM, 24-Jun-1993.)
⊢ Ⅎ𝑥𝜓    ⇒   ⊢ (∃𝑥(𝜑 → 𝜓) ↔ (∀𝑥𝜑 → 𝜓))
Theorem	19.36i 2239	Inference associated with 19.36 2238. See 19.36iv 1948 for a version requiring fewer axioms. (Contributed by NM, 24-Jun-1993.)
⊢ Ⅎ𝑥𝜓    &   ⊢ ∃𝑥(𝜑 → 𝜓)    ⇒   ⊢ (∀𝑥𝜑 → 𝜓)
Theorem	19.37 2240	Theorem 19.37 of [Margaris] p. 90. See 19.37v 1999 for a version requiring fewer axioms. (Contributed by NM, 21-Jun-1993.)
⊢ Ⅎ𝑥𝜑    ⇒   ⊢ (∃𝑥(𝜑 → 𝜓) ↔ (𝜑 → ∃𝑥𝜓))
Theorem	19.32 2241	Theorem 19.32 of [Margaris] p. 90. See 19.32v 1942 for a version requiring fewer axioms. (Contributed by NM, 14-May-1993.) (Revised by Mario Carneiro, 24-Sep-2016.)
⊢ Ⅎ𝑥𝜑    ⇒   ⊢ (∀𝑥(𝜑 ∨ 𝜓) ↔ (𝜑 ∨ ∀𝑥𝜓))
Theorem	19.31 2242	Theorem 19.31 of [Margaris] p. 90. See 19.31v 1943 for a version requiring fewer axioms. (Contributed by NM, 14-May-1993.)
⊢ Ⅎ𝑥𝜓    ⇒   ⊢ (∀𝑥(𝜑 ∨ 𝜓) ↔ (∀𝑥𝜑 ∨ 𝜓))
Theorem	19.41 2243	Theorem 19.41 of [Margaris] p. 90. See 19.41v 1951 for a version requiring fewer axioms. (Contributed by NM, 14-May-1993.) (Proof shortened by Andrew Salmon, 25-May-2011.) (Proof shortened by Wolf Lammen, 12-Jan-2018.)
⊢ Ⅎ𝑥𝜓    ⇒   ⊢ (∃𝑥(𝜑 ∧ 𝜓) ↔ (∃𝑥𝜑 ∧ 𝜓))
Theorem	19.42 2244	Theorem 19.42 of [Margaris] p. 90. See 19.42v 1955 for a version requiring fewer axioms. See exan 1864 for an immediate version. (Contributed by NM, 18-Aug-1993.)
⊢ Ⅎ𝑥𝜑    ⇒   ⊢ (∃𝑥(𝜑 ∧ 𝜓) ↔ (𝜑 ∧ ∃𝑥𝜓))
Theorem	19.44 2245	Theorem 19.44 of [Margaris] p. 90. See 19.44v 2000 for a version requiring fewer axioms. (Contributed by NM, 12-Mar-1993.)
⊢ Ⅎ𝑥𝜓    ⇒   ⊢ (∃𝑥(𝜑 ∨ 𝜓) ↔ (∃𝑥𝜑 ∨ 𝜓))
Theorem	19.45 2246	Theorem 19.45 of [Margaris] p. 90. See 19.45v 2001 for a version requiring fewer axioms. (Contributed by NM, 12-Mar-1993.)
⊢ Ⅎ𝑥𝜑    ⇒   ⊢ (∃𝑥(𝜑 ∨ 𝜓) ↔ (𝜑 ∨ ∃𝑥𝜓))
Theorem	spimfv 2247*	Specialization, using implicit substitution. Version of spim 2392 with a disjoint variable condition, which does not require ax-13 2377. See spimvw 1988 for a version with two disjoint variable conditions, requiring fewer axioms, and spimv 2395 for another variant. (Contributed by NM, 10-Jan-1993.) (Revised by BJ, 31-May-2019.)
⊢ Ⅎ𝑥𝜓    &   ⊢ (𝑥 = 𝑦 → (𝜑 → 𝜓))    ⇒   ⊢ (∀𝑥𝜑 → 𝜓)
Theorem	chvarfv 2248*	Implicit substitution of 𝑦 for 𝑥 into a theorem. Version of chvar 2400 with a disjoint variable condition, which does not require ax-13 2377. (Contributed by Raph Levien, 9-Jul-2003.) (Revised by BJ, 31-May-2019.)
⊢ Ⅎ𝑥𝜓    &   ⊢ (𝑥 = 𝑦 → (𝜑 ↔ 𝜓))    &   ⊢ 𝜑    ⇒   ⊢ 𝜓
Theorem	cbv3v2 2249*	Version of cbv3 2402 with two disjoint variable conditions, which does not require ax-11 2163 nor ax-13 2377. (Contributed by BJ, 24-Jun-2019.) (Proof shortened by Wolf Lammen, 30-Aug-2021.)
⊢ Ⅎ𝑥𝜓    &   ⊢ (𝑥 = 𝑦 → (𝜑 → 𝜓))    ⇒   ⊢ (∀𝑥𝜑 → ∀𝑦𝜓)
Theorem	sbalex 2250*	Equivalence of two ways to express proper substitution of a setvar for another setvar disjoint from it in a formula. This proof of their equivalence does not use df-sb 2069. That both sides of the biconditional express proper substitution is proved by sb5 2283 and sb6 2091. The implication "to the left" is equs4v 2002 and does not require ax-10 2147 nor ax-12 2185. It also holds without disjoint variable condition if we allow more axioms (see equs4 2421). Theorem 6.2 of [Quine] p. 40. Theorem equs5 2465 replaces the disjoint variable condition with a distinctor antecedent. Theorem equs45f 2464 replaces the disjoint variable condition on 𝑥, 𝑡 with the nonfreeness hypothesis of 𝑡 in 𝜑. (Contributed by NM, 14-Apr-2008.) Revised to use equsexv 2276 in place of equsex 2423 in order to remove dependency on ax-13 2377. (Revised by BJ, 20-Dec-2020.) Revise to remove dependency on df-sb 2069. (Revised by BJ, 21-Sep-2024.) (Proof shortened by SN, 14-Aug-2025.)
⊢ (∃𝑥(𝑥 = 𝑡 ∧ 𝜑) ↔ ∀𝑥(𝑥 = 𝑡 → 𝜑))
Theorem	sbalexOLD 2251*	Obsolete version of sbalex 2250 as of 14-Aug-2025. (Contributed by NM, 14-Apr-2008.) (Revised by BJ, 20-Dec-2020.) (Revised by BJ, 21-Sep-2024.) (Proof modification is discouraged.) (New usage is discouraged.)
⊢ (∃𝑥(𝑥 = 𝑡 ∧ 𝜑) ↔ ∀𝑥(𝑥 = 𝑡 → 𝜑))
Theorem	sb4av 2252*	Version of sb4a 2485 with a disjoint variable condition, which does not require ax-13 2377. The distinctor antecedent from sb4b 2480 is replaced by a disjoint variable condition in this theorem. (Contributed by NM, 2-Feb-2007.) (Revised by BJ, 15-Dec-2023.)
⊢ ([𝑡 / 𝑥]∀𝑡𝜑 → ∀𝑥(𝑥 = 𝑡 → 𝜑))
Theorem	sbimd 2253	Deduction substituting both sides of an implication. (Contributed by Wolf Lammen, 24-Nov-2022.) Revise df-sb 2069. (Revised by Steven Nguyen, 9-Jul-2023.)
⊢ Ⅎ𝑥𝜑    &   ⊢ (𝜑 → (𝜓 → 𝜒))    ⇒   ⊢ (𝜑 → ([𝑦 / 𝑥]𝜓 → [𝑦 / 𝑥]𝜒))
Theorem	sbbid 2254	Deduction substituting both sides of a biconditional. (Contributed by NM, 30-Jun-1993.) Remove dependency on ax-10 2147 and ax-13 2377. (Revised by Wolf Lammen, 24-Nov-2022.) Revise df-sb 2069. (Revised by Steven Nguyen, 11-Jul-2023.)
⊢ Ⅎ𝑥𝜑    &   ⊢ (𝜑 → (𝜓 ↔ 𝜒))    ⇒   ⊢ (𝜑 → ([𝑦 / 𝑥]𝜓 ↔ [𝑦 / 𝑥]𝜒))
Theorem	2sbbid 2255	Deduction doubly substituting both sides of a biconditional. (Contributed by AV, 30-Jul-2023.)
⊢ Ⅎ𝑥𝜑    &   ⊢ (𝜑 → (𝜓 ↔ 𝜒))    &   ⊢ Ⅎ𝑦𝜑    ⇒   ⊢ (𝜑 → ([𝑡 / 𝑥][𝑢 / 𝑦]𝜓 ↔ [𝑡 / 𝑥][𝑢 / 𝑦]𝜒))
Theorem	sbequ1 2256	An equality theorem for substitution. (Contributed by NM, 16-May-1993.) Revise df-sb 2069. (Revised by BJ, 22-Dec-2020.)
⊢ (𝑥 = 𝑡 → (𝜑 → [𝑡 / 𝑥]𝜑))
Theorem	sbequ2 2257	An equality theorem for substitution. (Contributed by NM, 16-May-1993.) Revise df-sb 2069. (Revised by BJ, 22-Dec-2020.) (Proof shortened by Wolf Lammen, 3-Feb-2024.)
⊢ (𝑥 = 𝑡 → ([𝑡 / 𝑥]𝜑 → 𝜑))
Theorem	stdpc7 2258	One of the two equality axioms of standard predicate calculus, called substitutivity of equality. (The other one is stdpc6 2030.) Translated to traditional notation, it can be read: "𝑥 = 𝑦 → (𝜑(𝑥, 𝑥) → 𝜑(𝑥, 𝑦)), provided that 𝑦 is free for 𝑥 in 𝜑(𝑥, 𝑥)". Axiom 7 of [Mendelson] p. 95. (Contributed by NM, 15-Feb-2005.)
⊢ (𝑥 = 𝑦 → ([𝑥 / 𝑦]𝜑 → 𝜑))
Theorem	sbequ12 2259	An equality theorem for substitution. (Contributed by NM, 14-May-1993.)
⊢ (𝑥 = 𝑦 → (𝜑 ↔ [𝑦 / 𝑥]𝜑))
Theorem	sbequ12r 2260	An equality theorem for substitution. (Contributed by NM, 6-Oct-2004.) (Proof shortened by Andrew Salmon, 21-Jun-2011.)
⊢ (𝑥 = 𝑦 → ([𝑥 / 𝑦]𝜑 ↔ 𝜑))
Theorem	sbelx 2261*	Elimination of substitution. Also see sbel2x 2479. (Contributed by NM, 5-Aug-1993.) Avoid ax-13 2377. (Revised by Wolf Lammen, 6-Aug-2023.) Avoid ax-10 2147. (Revised by GG, 20-Aug-2023.)
⊢ (𝜑 ↔ ∃𝑥(𝑥 = 𝑦 ∧ [𝑥 / 𝑦]𝜑))
Theorem	sbequ12a 2262	An equality theorem for substitution. (Contributed by NM, 2-Jun-1993.) (Proof shortened by Wolf Lammen, 23-Jun-2019.)
⊢ (𝑥 = 𝑦 → ([𝑦 / 𝑥]𝜑 ↔ [𝑥 / 𝑦]𝜑))
Theorem	sbid 2263	An identity theorem for substitution. Remark 9.1 in [Megill] p. 447 (p. 15 of the preprint). (Contributed by NM, 26-May-1993.) (Proof shortened by Wolf Lammen, 30-Sep-2018.)
⊢ ([𝑥 / 𝑥]𝜑 ↔ 𝜑)
Theorem	sbcov 2264*	A composition law for substitution. Version of sbco 2512 with a disjoint variable condition using fewer axioms. (Contributed by NM, 14-May-1993.) (Revised by GG, 7-Aug-2023.) (Proof shortened by SN, 26-Aug-2025.)
⊢ ([𝑦 / 𝑥][𝑥 / 𝑦]𝜑 ↔ [𝑦 / 𝑥]𝜑)
Theorem	sbcovOLD 2265*	Obsolete version of sbcov 2264 as of 26-Aug-2025. (Contributed by NM, 14-May-1993.) (Revised by GG, 7-Aug-2023.) (Proof modification is discouraged.) (New usage is discouraged.)
⊢ ([𝑦 / 𝑥][𝑥 / 𝑦]𝜑 ↔ [𝑦 / 𝑥]𝜑)
Theorem	sb6a 2266*	Equivalence for substitution. (Contributed by NM, 2-Jun-1993.) (Proof shortened by Wolf Lammen, 23-Sep-2018.)
⊢ ([𝑦 / 𝑥]𝜑 ↔ ∀𝑥(𝑥 = 𝑦 → [𝑥 / 𝑦]𝜑))
Theorem	sbid2vw 2267*	Reverting substitution yields the original expression. Based on fewer axioms than sbid2v 2514, at the expense of an extra distinct variable condition. (Contributed by NM, 14-May-1993.) (Revised by Wolf Lammen, 5-Aug-2023.)
⊢ ([𝑡 / 𝑥][𝑥 / 𝑡]𝜑 ↔ 𝜑)
Theorem	axc16g 2268*	Generalization of axc16 2269. Use the latter when sufficient. This proof only requires, on top of { ax-1 6-- ax-7 2010 }, Theorem ax12v 2186. (Contributed by NM, 15-May-1993.) (Proof shortened by Andrew Salmon, 25-May-2011.) (Proof shortened by Wolf Lammen, 18-Feb-2018.) Remove dependency on ax-13 2377, along an idea of BJ. (Revised by Wolf Lammen, 30-Nov-2019.) (Revised by BJ, 7-Jul-2021.) Shorten axc11rv 2273. (Revised by Wolf Lammen, 11-Oct-2021.)
⊢ (∀𝑥 𝑥 = 𝑦 → (𝜑 → ∀𝑧𝜑))
Theorem	axc16 2269*	Proof of older axiom ax-c16 39189. (Contributed by NM, 8-Nov-2006.) (Revised by NM, 22-Sep-2017.)
⊢ (∀𝑥 𝑥 = 𝑦 → (𝜑 → ∀𝑥𝜑))
Theorem	axc16gb 2270*	Biconditional strengthening of axc16g 2268. (Contributed by NM, 15-May-1993.)
⊢ (∀𝑥 𝑥 = 𝑦 → (𝜑 ↔ ∀𝑧𝜑))
Theorem	axc16nf 2271*	If dtru 5387 is false, then there is only one element in the universe, so everything satisfies Ⅎ. (Contributed by Mario Carneiro, 7-Oct-2016.) Remove dependency on ax-11 2163. (Revised by Wolf Lammen, 9-Sep-2018.) (Proof shortened by BJ, 14-Jun-2019.) Remove dependency on ax-10 2147. (Revised by Wolf Lammen, 12-Oct-2021.)
⊢ (∀𝑥 𝑥 = 𝑦 → Ⅎ𝑧𝜑)
Theorem	axc11v 2272*	Version of axc11 2435 with a disjoint variable condition on 𝑥 and 𝑦, which is provable, on top of { ax-1 6-- ax-7 2010 }, from ax12v 2186 (contrary to axc11 2435 which seems to require the full ax-12 2185 and ax-13 2377). (Contributed by NM, 16-May-2008.) (Revised by BJ, 6-Jul-2021.) (Proof shortened by Wolf Lammen, 11-Oct-2021.)
⊢ (∀𝑥 𝑥 = 𝑦 → (∀𝑥𝜑 → ∀𝑦𝜑))
Theorem	axc11rv 2273*	Version of axc11r 2373 with a disjoint variable condition on 𝑥 and 𝑦, which is provable, on top of { ax-1 6-- ax-7 2010 }, from ax12v 2186 (contrary to axc11 2435 which seems to require the full ax-12 2185 and ax-13 2377, and to axc11r 2373 which seems to require the full ax-12 2185). (Contributed by BJ, 6-Jul-2021.) (Proof shortened by Wolf Lammen, 11-Oct-2021.)
⊢ (∀𝑥 𝑥 = 𝑦 → (∀𝑦𝜑 → ∀𝑥𝜑))
Theorem	drsb2 2274	Formula-building lemma for use with the Distinctor Reduction Theorem. Part of Theorem 9.4 of [Megill] p. 448 (p. 16 of preprint). (Contributed by NM, 27-Feb-2005.)
⊢ (∀𝑥 𝑥 = 𝑦 → ([𝑥 / 𝑧]𝜑 ↔ [𝑦 / 𝑧]𝜑))
Theorem	equsalv 2275*	An equivalence related to implicit substitution. Version of equsal 2422 with a disjoint variable condition, which does not require ax-13 2377. See equsalvw 2006 for a version with two disjoint variable conditions requiring fewer axioms. See also the dual form equsexv 2276. (Contributed by NM, 2-Jun-1993.) (Revised by BJ, 31-May-2019.)
⊢ Ⅎ𝑥𝜓    &   ⊢ (𝑥 = 𝑦 → (𝜑 ↔ 𝜓))    ⇒   ⊢ (∀𝑥(𝑥 = 𝑦 → 𝜑) ↔ 𝜓)
Theorem	equsexv 2276*	An equivalence related to implicit substitution. Version of equsex 2423 with a disjoint variable condition, which does not require ax-13 2377. See equsexvw 2007 for a version with two disjoint variable conditions requiring fewer axioms. See also the dual form equsalv 2275. (Contributed by NM, 5-Aug-1993.) (Revised by BJ, 31-May-2019.) Avoid ax-10 2147. (Revised by GG, 18-Nov-2024.)
⊢ Ⅎ𝑥𝜓    &   ⊢ (𝑥 = 𝑦 → (𝜑 ↔ 𝜓))    ⇒   ⊢ (∃𝑥(𝑥 = 𝑦 ∧ 𝜑) ↔ 𝜓)
Theorem	sbft 2277	Substitution has no effect on a nonfree variable. (Contributed by NM, 30-May-2009.) (Revised by Mario Carneiro, 12-Oct-2016.) (Proof shortened by Wolf Lammen, 3-May-2018.)
⊢ (Ⅎ𝑥𝜑 → ([𝑦 / 𝑥]𝜑 ↔ 𝜑))
Theorem	sbf 2278	Substitution for a variable not free in a wff does not affect it. For a version requiring disjoint variables but fewer axioms, see sbv 2094. (Contributed by NM, 14-May-1993.) (Revised by Mario Carneiro, 4-Oct-2016.)
⊢ Ⅎ𝑥𝜑    ⇒   ⊢ ([𝑦 / 𝑥]𝜑 ↔ 𝜑)
Theorem	sbf2 2279	Substitution has no effect on a bound variable. (Contributed by NM, 1-Jul-2005.)
⊢ ([𝑦 / 𝑥]∀𝑥𝜑 ↔ ∀𝑥𝜑)
Theorem	sbh 2280	Substitution for a variable not free in a wff does not affect it. (Contributed by NM, 14-May-1993.)
⊢ (𝜑 → ∀𝑥𝜑)    ⇒   ⊢ ([𝑦 / 𝑥]𝜑 ↔ 𝜑)
Theorem	hbs1 2281*	The setvar 𝑥 is not free in [𝑦 / 𝑥]𝜑 when 𝑥 and 𝑦 are distinct. (Contributed by NM, 26-May-1993.)
⊢ ([𝑦 / 𝑥]𝜑 → ∀𝑥[𝑦 / 𝑥]𝜑)
Theorem	nfs1f 2282	If 𝑥 is not free in 𝜑, it is not free in [𝑦 / 𝑥]𝜑. (Contributed by Mario Carneiro, 11-Aug-2016.)
⊢ Ⅎ𝑥𝜑    ⇒   ⊢ Ⅎ𝑥[𝑦 / 𝑥]𝜑
Theorem	sb5 2283*	Alternate definition of substitution when variables are disjoint. Similar to Theorem 6.1 of [Quine] p. 40. The implication "to the right" is sb1v 2093 and even needs no disjoint variable condition, see sb1 2483. Theorem sb5f 2503 replaces the disjoint variable condition with a nonfreeness hypothesis. (Contributed by NM, 18-Aug-1993.) (Revised by Wolf Lammen, 4-Sep-2023.)
⊢ ([𝑦 / 𝑥]𝜑 ↔ ∃𝑥(𝑥 = 𝑦 ∧ 𝜑))
Theorem	equs5av 2284*	A property related to substitution that replaces the distinctor from equs5 2465 to a disjoint variable condition. Version of equs5a 2462 with a disjoint variable condition, which does not require ax-13 2377. See also sbalex 2250. (Contributed by NM, 2-Feb-2007.) (Revised by GG, 15-Dec-2023.)
⊢ (∃𝑥(𝑥 = 𝑦 ∧ ∀𝑦𝜑) → ∀𝑥(𝑥 = 𝑦 → 𝜑))
Theorem	2sb5 2285*	Equivalence for double substitution. (Contributed by NM, 3-Feb-2005.)
⊢ ([𝑧 / 𝑥][𝑤 / 𝑦]𝜑 ↔ ∃𝑥∃𝑦((𝑥 = 𝑧 ∧ 𝑦 = 𝑤) ∧ 𝜑))
Theorem	dfsb7 2286*	An alternate definition of proper substitution df-sb 2069. By introducing a dummy variable 𝑦 in the definiens, we are able to eliminate any distinct variable restrictions among the variables 𝑡, 𝑥, and 𝜑 of the definiendum. No distinct variable conflicts arise because 𝑦 effectively insulates 𝑡 from 𝑥. To achieve this, we use a chain of two substitutions in the form of sb5 2283, first 𝑦 for 𝑥 then 𝑡 for 𝑦. Compare Definition 2.1'' of [Quine] p. 17, which is obtained from this theorem by applying df-clab 2716. Theorem sb7h 2531 provides a version where 𝜑 and 𝑦 don't have to be distinct. (Contributed by NM, 28-Jan-2004.) Revise df-sb 2069. (Revised by BJ, 25-Dec-2020.) (Proof shortened by Wolf Lammen, 3-Sep-2023.)
⊢ ([𝑡 / 𝑥]𝜑 ↔ ∃𝑦(𝑦 = 𝑡 ∧ ∃𝑥(𝑥 = 𝑦 ∧ 𝜑)))
Theorem	sbn 2287	Negation inside and outside of substitution are equivalent. (Contributed by NM, 14-May-1993.) (Proof shortened by Wolf Lammen, 30-Apr-2018.) Revise df-sb 2069. (Revised by BJ, 25-Dec-2020.)
⊢ ([𝑡 / 𝑥] ¬ 𝜑 ↔ ¬ [𝑡 / 𝑥]𝜑)
Theorem	sbex 2288*	Move existential quantifier in and out of substitution. (Contributed by NM, 27-Sep-2003.)
⊢ ([𝑧 / 𝑦]∃𝑥𝜑 ↔ ∃𝑥[𝑧 / 𝑦]𝜑)
Theorem	nf5 2289	Alternate definition of df-nf 1786. (Contributed by Mario Carneiro, 11-Aug-2016.) df-nf 1786 changed. (Revised by Wolf Lammen, 11-Sep-2021.)
⊢ (Ⅎ𝑥𝜑 ↔ ∀𝑥(𝜑 → ∀𝑥𝜑))
Theorem	nf6 2290	An alternate definition of df-nf 1786. (Contributed by Mario Carneiro, 24-Sep-2016.)
⊢ (Ⅎ𝑥𝜑 ↔ ∀𝑥(∃𝑥𝜑 → 𝜑))
Theorem	nf5d 2291	Deduce that 𝑥 is not free in 𝜓 in a context. (Contributed by Mario Carneiro, 24-Sep-2016.)
⊢ Ⅎ𝑥𝜑    &   ⊢ (𝜑 → (𝜓 → ∀𝑥𝜓))    ⇒   ⊢ (𝜑 → Ⅎ𝑥𝜓)
Theorem	nf5di 2292	Since the converse holds by a1i 11, this inference shows that we can represent a not-free hypothesis with either Ⅎ𝑥𝜑 (inference form) or (𝜑 → Ⅎ𝑥𝜑) (deduction form). (Contributed by NM, 17-Aug-2018.) (Proof shortened by Wolf Lammen, 10-Jul-2019.)
⊢ (𝜑 → Ⅎ𝑥𝜑)    ⇒   ⊢ Ⅎ𝑥𝜑
Theorem	19.9h 2293	A wff may be existentially quantified with a variable not free in it. Theorem 19.9 of [Margaris] p. 89. (Contributed by FL, 24-Mar-2007.) (Proof shortened by Wolf Lammen, 5-Jan-2018.) (Proof shortened by Wolf Lammen, 14-Jul-2020.)
⊢ (𝜑 → ∀𝑥𝜑)    ⇒   ⊢ (∃𝑥𝜑 ↔ 𝜑)
Theorem	19.21h 2294	Theorem 19.21 of [Margaris] p. 90. The hypothesis can be thought of as "𝑥 is not free in 𝜑". See also 19.21 2215 and 19.21v 1941. (Contributed by NM, 1-Aug-2017.) (Proof shortened by Wolf Lammen, 1-Jan-2018.)
⊢ (𝜑 → ∀𝑥𝜑)    ⇒   ⊢ (∀𝑥(𝜑 → 𝜓) ↔ (𝜑 → ∀𝑥𝜓))
Theorem	19.23h 2295	Theorem 19.23 of [Margaris] p. 90. See 19.23 2219. (Contributed by NM, 24-Jan-1993.) (Revised by Mario Carneiro, 24-Sep-2016.) (Proof shortened by Wolf Lammen, 1-Jan-2018.)
⊢ (𝜓 → ∀𝑥𝜓)    ⇒   ⊢ (∀𝑥(𝜑 → 𝜓) ↔ (∃𝑥𝜑 → 𝜓))
Theorem	exlimih 2296	Inference associated with 19.23 2219. See exlimiv 1932 for a version with a disjoint variable condition requiring fewer axioms. (Contributed by NM, 10-Jan-1993.) (Proof shortened by Andrew Salmon, 13-May-2011.) (Proof shortened by Wolf Lammen, 1-Jan-2018.)
⊢ (𝜓 → ∀𝑥𝜓)    &   ⊢ (𝜑 → 𝜓)    ⇒   ⊢ (∃𝑥𝜑 → 𝜓)
Theorem	exlimdh 2297	Deduction form of Theorem 19.9 of [Margaris] p. 89. (Contributed by NM, 28-Jan-1997.)
⊢ (𝜑 → ∀𝑥𝜑)    &   ⊢ (𝜒 → ∀𝑥𝜒)    &   ⊢ (𝜑 → (𝜓 → 𝜒))    ⇒   ⊢ (𝜑 → (∃𝑥𝜓 → 𝜒))
Theorem	equsalhw 2298*	Version of equsalh 2425 with a disjoint variable condition, which does not require ax-13 2377. (Contributed by NM, 29-Nov-2015.) (Proof shortened by Wolf Lammen, 8-Jul-2022.)
⊢ (𝜓 → ∀𝑥𝜓)    &   ⊢ (𝑥 = 𝑦 → (𝜑 ↔ 𝜓))    ⇒   ⊢ (∀𝑥(𝑥 = 𝑦 → 𝜑) ↔ 𝜓)
Theorem	equsexhv 2299*	An equivalence related to implicit substitution. Version of equsexh 2426 with a disjoint variable condition, which does not require ax-13 2377. (Contributed by NM, 5-Aug-1993.) (Revised by BJ, 31-May-2019.)
⊢ (𝜓 → ∀𝑥𝜓)    &   ⊢ (𝑥 = 𝑦 → (𝜑 ↔ 𝜓))    ⇒   ⊢ (∃𝑥(𝑥 = 𝑦 ∧ 𝜑) ↔ 𝜓)
Theorem	hba1 2300	The setvar 𝑥 is not free in ∀𝑥𝜑. This corresponds to the axiom (4) of modal logic. Example in Appendix in [Megill] p. 450 (p. 19 of the preprint). Also Lemma 22 of [Monk2] p. 114. (Contributed by NM, 24-Jan-1993.) (Proof shortened by Wolf Lammen, 12-Oct-2021.)
⊢ (∀𝑥𝜑 → ∀𝑥∀𝑥𝜑)