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	excom13 2201	Swap 1st and 3rd existential quantifiers. (Contributed by NM, 9-Mar-1995.)
⊢ (∃𝑥∃𝑦∃𝑧𝜑 ↔ ∃𝑧∃𝑦∃𝑥𝜑)
Theorem	exrot3 2202	Rotate existential quantifiers. (Contributed by NM, 17-Mar-1995.)
⊢ (∃𝑥∃𝑦∃𝑧𝜑 ↔ ∃𝑦∃𝑧∃𝑥𝜑)
Theorem	exrot4 2203	Rotate existential quantifiers twice. (Contributed by NM, 9-Mar-1995.)
⊢ (∃𝑥∃𝑦∃𝑧∃𝑤𝜑 ↔ ∃𝑧∃𝑤∃𝑥∃𝑦𝜑)
Theorem	hbal 2204	If 𝑥 is not free in 𝜑, it is not free in ∀𝑦𝜑. (Contributed by NM, 12-Mar-1993.)
⊢ (𝜑 → ∀𝑥𝜑)    ⇒   ⊢ (∀𝑦𝜑 → ∀𝑥∀𝑦𝜑)
Theorem	hbald 2205	Deduction form of bound-variable hypothesis builder hbal 2204. (Contributed by NM, 2-Jan-2002.)
⊢ (𝜑 → ∀𝑦𝜑)    &   ⊢ (𝜑 → (𝜓 → ∀𝑥𝜓))    ⇒   ⊢ (𝜑 → (∀𝑦𝜓 → ∀𝑥∀𝑦𝜓))
Theorem	sbal 2206*	Move universal quantifier in and out of substitution. (Contributed by NM, 16-May-1993.) (Proof shortened by Wolf Lammen, 29-Sep-2018.) Reduce dependencies on axioms. (Revised by Steven Nguyen, 13-Aug-2023.)
⊢ ([𝑧 / 𝑦]∀𝑥𝜑 ↔ ∀𝑥[𝑧 / 𝑦]𝜑)
Theorem	sbalv 2207*	Quantify with new variable inside substitution. (Contributed by NM, 18-Aug-1993.)
⊢ ([𝑦 / 𝑥]𝜑 ↔ 𝜓)    ⇒   ⊢ ([𝑦 / 𝑥]∀𝑧𝜑 ↔ ∀𝑧𝜓)
Theorem	hbsbw 2208*	If 𝑧 is not free in 𝜑, it is not free in [𝑦 / 𝑥]𝜑 when 𝑦 and 𝑧 are distinct. Version of hbsb 2558 with a disjoint variable condition, which requires fewer axioms. (Contributed by NM, 12-Aug-1993.) Remove dependencies on axioms. (Revised by GG, 23-May-2024.) (Proof shortened by Wolf Lammen, 14-May-2025.)
⊢ (𝜑 → ∀𝑧𝜑)    ⇒   ⊢ ([𝑦 / 𝑥]𝜑 → ∀𝑧[𝑦 / 𝑥]𝜑)
Theorem	sbcom2 2209*	Commutativity law for substitution. Used in proof of Theorem 9.7 of [Megill] p. 449 (p. 16 of the preprint). (Contributed by NM, 27-May-1997.) (Proof shortened by Wolf Lammen, 23-Dec-2022.)
⊢ ([𝑤 / 𝑧][𝑦 / 𝑥]𝜑 ↔ [𝑦 / 𝑥][𝑤 / 𝑧]𝜑)
Theorem	sbco4lemOLD 2210*	Obsolete version of sbco4lem 2138 as of 3-Sep-2025. (Contributed by Jim Kingdon, 26-Sep-2018.) (Proof shortened by Wolf Lammen, 12-Oct-2024.) (Proof modification is discouraged.) (New usage is discouraged.)
⊢ ([𝑥 / 𝑣][𝑦 / 𝑥][𝑣 / 𝑦]𝜑 ↔ [𝑥 / 𝑤][𝑦 / 𝑥][𝑤 / 𝑦]𝜑)
Theorem	sbco4OLD 2211*	Obsolete version of sbco4 2139 as of 3-Sep-2025. (Contributed by Jim Kingdon, 25-Sep-2018.) (Proof modification is discouraged.) (New usage is discouraged.)
⊢ ([𝑦 / 𝑢][𝑥 / 𝑣][𝑢 / 𝑥][𝑣 / 𝑦]𝜑 ↔ [𝑥 / 𝑤][𝑦 / 𝑥][𝑤 / 𝑦]𝜑)
Theorem	nfa2 2212	Lemma 24 of [Monk2] p. 114. (Contributed by Mario Carneiro, 24-Sep-2016.) Remove dependency on ax-12 2215. (Revised by Wolf Lammen, 18-Oct-2021.)
⊢ Ⅎ𝑥∀𝑦∀𝑥𝜑
Theorem	nfexhe 2213	Version of nfex 2359 with the existential dual to the 'h' hypothesis, avoiding ax-12 2215. (Contributed by SN, 11-Feb-2026.)
⊢ (∃𝑥𝜑 → 𝜑)    ⇒   ⊢ Ⅎ𝑥∃𝑦𝜑
Theorem	nfexa2 2214	An inner universal quantifier's variable is bound. (Contributed by SN, 11-Feb-2026.)
⊢ Ⅎ𝑥∃𝑦∀𝑥𝜑
1.5.3  Axiom scheme ax-12 (Substitution)
Axiom	ax-12 2215	Axiom of Substitution. One of the 5 equality axioms of predicate calculus. The final consequent ∀𝑥(𝑥 = 𝑦 → 𝜑) is a way of expressing "𝑦 substituted for 𝑥 in wff 𝜑 " (cf. sb6 2121). It is based on Lemma 16 of [Tarski] p. 70 and Axiom C8 of [Monk2] p. 105, from which it can be proved by cases. The original version of this axiom was ax-c15 39525 and was replaced with this shorter ax-12 2215 in Jan. 2007. The old axiom is proved from this one as Theorem axc15 2456. Conversely, this axiom is proved from ax-c15 39525 as Theorem ax12 2457. Juha Arpiainen proved the metalogical independence of this axiom (in the form of the older axiom ax-c15 39525) from the others on 19-Jan-2006. See item 9a at https://us.metamath.org/award2003.html 39525. See ax12v 2216 and ax12v2 2217 for other equivalents of this axiom that (unlike this axiom) have distinct variable restrictions. This axiom scheme is logically redundant (see ax12w 2170) but is used as an auxiliary axiom scheme to achieve scheme completeness. (Contributed by NM, 22-Jan-2007.) (New usage is discouraged.)
⊢ (𝑥 = 𝑦 → (∀𝑦𝜑 → ∀𝑥(𝑥 = 𝑦 → 𝜑)))
Theorem	ax12v 2216*	This is essentially Axiom ax-12 2215 weakened by additional restrictions on variables. Besides axc11r 2402, this theorem should be the only one referencing ax-12 2215 directly. Both restrictions on variables have their own value. If for a moment we assume 𝑥 could be set to 𝑦, then, after elimination of the tautology 𝑦 = 𝑦, immediately we have 𝜑 → ∀𝑦𝜑 for all 𝜑 and 𝑦, that is ax-5 1933, a degenerate result. The second restriction is not necessary, but a simplification that makes the following interpretation easier to see. Since 𝜑 textually at most depends on 𝑥, we can look at it at some given 'fixed' 𝑦. This theorem now states that the truth value of 𝜑 will stay constant, as long as we 'vary 𝑥 around 𝑦' only such that 𝑥 = 𝑦 still holds. Or in other words, equality is the finest grained logical expression. If you cannot differ two sets by =, you won't find a whatever sophisticated expression that does. One might wonder how the described variation of 𝑥 is possible at all. Note that Metamath is a text processor that easily sees a difference between text chunks {𝑥 ∣ ¬ 𝑥 = 𝑥} and {𝑦 ∣ ¬ 𝑦 = 𝑦}. Our usual interpretation is to abstract from textual variations of the same set, but we are free to interpret Metamath's formalism differently, and in fact let 𝑥 run through all textual representations of sets. Had we allowed 𝜑 to depend also on 𝑦, this idea is both harder to see, and it is less clear that this extra freedom introduces effects not covered by other axioms. (Contributed by Wolf Lammen, 8-Aug-2020.)
⊢ (𝑥 = 𝑦 → (𝜑 → ∀𝑥(𝑥 = 𝑦 → 𝜑)))
Theorem	ax12v2 2217*	It is possible to remove any restriction on 𝜑 in ax12v 2216. Same as Axiom C8 of [Monk2] p. 105. Use ax12v 2216 instead when sufficient. (Contributed by NM, 5-Aug-1993.) Remove dependencies on ax-10 2178 and ax-13 2406. (Revised by Jim Kingdon, 15-Dec-2017.) (Proof shortened by Wolf Lammen, 8-Dec-2019.)
⊢ (𝑥 = 𝑦 → (𝜑 → ∀𝑥(𝑥 = 𝑦 → 𝜑)))
Theorem	ax12ev2 2218*	Version of ax12v2 2217 rewritten to use an existential quantifier. One direction of sbalex 2280 without the universal quantifier, avoiding ax-10 2178. (Contributed by SN, 14-Aug-2025.)
⊢ (∃𝑥(𝑥 = 𝑦 ∧ 𝜑) → (𝑥 = 𝑦 → 𝜑))
Theorem	19.8a 2219	If a wff is true, it is true for at least one instance. Special case of Theorem 19.8 of [Margaris] p. 89. See 19.8v 2006 for a version with a disjoint variable condition requiring fewer axioms. (Contributed by NM, 9-Jan-1993.) Allow a shortening of sp 2221. (Revised by Wolf Lammen, 13-Jan-2018.) (Proof shortened by Wolf Lammen, 8-Dec-2019.)
⊢ (𝜑 → ∃𝑥𝜑)
Theorem	19.8ad 2220	If a wff is true, it is true for at least one instance. Deduction form of 19.8a 2219. (Contributed by DAW, 13-Feb-2017.)
⊢ (𝜑 → 𝜓)    ⇒   ⊢ (𝜑 → ∃𝑥𝜓)
Theorem	sp 2221	Specialization. A universally quantified wff implies the wff without a quantifier. Axiom scheme B5 of [Tarski] p. 67 (under his system S2, defined in the last paragraph on p. 77). Also appears as Axiom scheme C5' in [Megill] p. 448 (p. 16 of the preprint). This corresponds to the axiom (T) of modal logic. For the axiom of specialization presented in many logic textbooks, see Theorem stdpc4 2101. This theorem shows that our obsolete axiom ax-c5 39519 can be derived from the others. The proof uses ideas from the proof of Lemma 21 of [Monk2] p. 114. It appears that this scheme cannot be derived directly from Tarski's axioms without auxiliary axiom scheme ax-12 2215. It is thought the best we can do using only Tarski's axioms is spw 2057. Also see spvw 2004 where 𝑥 and 𝜑 are disjoint, using fewer axioms. (Contributed by NM, 21-May-2008.) (Proof shortened by Scott Fenton, 24-Jan-2011.) (Proof shortened by Wolf Lammen, 13-Jan-2018.)
⊢ (∀𝑥𝜑 → 𝜑)
Theorem	spi 2222	Inference rule of universal instantiation, or universal specialization. Converse of the inference rule of (universal) generalization ax-gen 1818. Contrary to the rule of generalization, its closed form is valid, see sp 2221. (Contributed by NM, 5-Aug-1993.)
⊢ ∀𝑥𝜑    ⇒   ⊢ 𝜑
Theorem	sps 2223	Generalization of antecedent. (Contributed by NM, 5-Jan-1993.)
⊢ (𝜑 → 𝜓)    ⇒   ⊢ (∀𝑥𝜑 → 𝜓)
Theorem	2sp 2224	A double specialization (see sp 2221). Another double specialization, closer to PM*11.1, is 2stdpc4 2104. (Contributed by BJ, 15-Sep-2018.)
⊢ (∀𝑥∀𝑦𝜑 → 𝜑)
Theorem	spsd 2225	Deduction generalizing antecedent. (Contributed by NM, 17-Aug-1994.)
⊢ (𝜑 → (𝜓 → 𝜒))    ⇒   ⊢ (𝜑 → (∀𝑥𝜓 → 𝜒))
Theorem	19.2g 2226	Theorem 19.2 of [Margaris] p. 89, generalized to use two setvar variables. Use 19.2 1999 when sufficient. (Contributed by Mel L. O'Cat, 31-Mar-2008.)
⊢ (∀𝑥𝜑 → ∃𝑦𝜑)
Theorem	19.21bi 2227	Inference form of 19.21 2245 and also deduction form of sp 2221. (Contributed by NM, 26-May-1993.)
⊢ (𝜑 → ∀𝑥𝜓)    ⇒   ⊢ (𝜑 → 𝜓)
Theorem	19.21bbi 2228	Inference removing two universal quantifiers. Version of 19.21bi 2227 with two quantifiers. (Contributed by NM, 20-Apr-1994.)
⊢ (𝜑 → ∀𝑥∀𝑦𝜓)    ⇒   ⊢ (𝜑 → 𝜓)
Theorem	19.23bi 2229	Inference form of Theorem 19.23 of [Margaris] p. 90, see 19.23 2249. (Contributed by NM, 12-Mar-1993.)
⊢ (∃𝑥𝜑 → 𝜓)    ⇒   ⊢ (𝜑 → 𝜓)
Theorem	nexr 2230	Inference associated with the contrapositive of 19.8a 2219. (Contributed by Jeff Hankins, 26-Jul-2009.)
⊢ ¬ ∃𝑥𝜑    ⇒   ⊢ ¬ 𝜑
Theorem	qexmid 2231	Quantified excluded middle (see exmid 907). 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 2232	Consequence of the definition of not-free. (Contributed by Mario Carneiro, 26-Sep-2016.) df-nf 1807 changed. (Revised by Wolf Lammen, 11-Sep-2021.) (Proof shortened by Wolf Lammen, 23-Nov-2023.)
⊢ (Ⅎ𝑥𝜑 → (𝜑 → ∀𝑥𝜑))
Theorem	nf5ri 2233	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 2234	Consequence of the definition of not-free in a context. (Contributed by Mario Carneiro, 11-Aug-2016.)
⊢ (𝜑 → Ⅎ𝑥𝜓)    ⇒   ⊢ (𝜑 → (𝜓 → ∀𝑥𝜓))
Theorem	spimedv 2235*	Deduction version of spimev 2426. Version of spimed 2422 with a disjoint variable condition, which does not require ax-13 2406. See spime 2423 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 2236*	Version of spime 2423 with a disjoint variable condition, which does not require ax-13 2406. (Contributed by BJ, 31-May-2019.)
⊢ Ⅎ𝑥𝜑    &   ⊢ (𝑥 = 𝑦 → (𝜑 → 𝜓))    ⇒   ⊢ (𝜑 → ∃𝑥𝜓)
Theorem	nfim1 2237	A closed form of nfim 1919. (Contributed by NM, 2-Jun-1993.) (Revised by Mario Carneiro, 24-Sep-2016.) (Proof shortened by Wolf Lammen, 2-Jan-2018.) df-nf 1807 changed. (Revised by Wolf Lammen, 18-Sep-2021.)
⊢ Ⅎ𝑥𝜑    &   ⊢ (𝜑 → Ⅎ𝑥𝜓)    ⇒   ⊢ Ⅎ𝑥(𝜑 → 𝜓)
Theorem	nfan1 2238	A closed form of nfan 1922. (Contributed by Mario Carneiro, 3-Oct-2016.) df-nf 1807 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 2239	Closed form of 19.3 2240 and version of 19.9t 2242 with a universal quantifier. (Contributed by NM, 9-Nov-2020.) (Proof shortened by BJ, 9-Oct-2022.)
⊢ (Ⅎ𝑥𝜑 → (∀𝑥𝜑 ↔ 𝜑))
Theorem	19.3 2240	A wff may be quantified with a variable not free in it. Version of 19.9 2243 with a universal quantifier. Theorem 19.3 of [Margaris] p. 89. See 19.3v 2005 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 2241	A deduction version of one direction of 19.9 2243. (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 1807 changed. (Revised by Wolf Lammen, 11-Sep-2021.) (Proof shortened by Wolf Lammen, 8-Jul-2022.)
⊢ (𝜓 → Ⅎ𝑥𝜑)    ⇒   ⊢ (𝜓 → (∃𝑥𝜑 → 𝜑))
Theorem	19.9t 2242	Closed form of 19.9 2243 and version of 19.3t 2239 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 2243	A wff may be existentially quantified with a variable not free in it. Version of 19.3 2240 with an existential quantifier. Theorem 19.9 of [Margaris] p. 89. See 19.9v 2007 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 2244	Closed form of Theorem 19.21 of [Margaris] p. 90, see 19.21 2245. (Contributed by NM, 27-May-1997.) (Revised by Mario Carneiro, 24-Sep-2016.) (Proof shortened by Wolf Lammen, 3-Jan-2018.) df-nf 1807 changed. (Revised by Wolf Lammen, 11-Sep-2021.) (Proof shortened by BJ, 3-Nov-2021.)
⊢ (Ⅎ𝑥𝜑 → (∀𝑥(𝜑 → 𝜓) ↔ (𝜑 → ∀𝑥𝜓)))
Theorem	19.21 2245	Theorem 19.21 of [Margaris] p. 90. The hypothesis can be thought of as "𝑥 is not free in 𝜑". See 19.21v 1962 for a version requiring fewer axioms. See also 19.21h 2324. (Contributed by NM, 14-May-1993.) (Revised by Mario Carneiro, 24-Sep-2016.) df-nf 1807 changed. (Revised by Wolf Lammen, 18-Sep-2021.)
⊢ Ⅎ𝑥𝜑    ⇒   ⊢ (∀𝑥(𝜑 → 𝜓) ↔ (𝜑 → ∀𝑥𝜓))
Theorem	stdpc5 2246	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 2036. 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 1961 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 2178. (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 2247	Version of 19.21 2245 with two quantifiers. (Contributed by NM, 4-Feb-2005.)
⊢ Ⅎ𝑥𝜑    &   ⊢ Ⅎ𝑦𝜑    ⇒   ⊢ (∀𝑥∀𝑦(𝜑 → 𝜓) ↔ (𝜑 → ∀𝑥∀𝑦𝜓))
Theorem	19.23t 2248	Closed form of Theorem 19.23 of [Margaris] p. 90. See 19.23 2249. (Contributed by NM, 7-Nov-2005.) (Proof shortened by Wolf Lammen, 13-Aug-2020.) df-nf 1807 changed. (Revised by Wolf Lammen, 11-Sep-2021.) (Proof shortened by BJ, 8-Oct-2022.)
⊢ (Ⅎ𝑥𝜓 → (∀𝑥(𝜑 → 𝜓) ↔ (∃𝑥𝜑 → 𝜓)))
Theorem	19.23 2249	Theorem 19.23 of [Margaris] p. 90. See 19.23v 1965 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 2250	Deduction form of Theorem 19.20 of [Margaris] p. 90, see alim 1833. See alimdh 1840, alimdv 1939 for variants requiring fewer axioms. (Contributed by Mario Carneiro, 24-Sep-2016.)
⊢ Ⅎ𝑥𝜑    &   ⊢ (𝜑 → (𝜓 → 𝜒))    ⇒   ⊢ (𝜑 → (∀𝑥𝜓 → ∀𝑥𝜒))
Theorem	alrimi 2251	Inference form of Theorem 19.21 of [Margaris] p. 90, see 19.21 2245. (Contributed by Mario Carneiro, 24-Sep-2016.)
⊢ Ⅎ𝑥𝜑    &   ⊢ (𝜑 → 𝜓)    ⇒   ⊢ (𝜑 → ∀𝑥𝜓)
Theorem	alrimdd 2252	Deduction form of Theorem 19.21 of [Margaris] p. 90, see 19.21 2245. (Contributed by Mario Carneiro, 24-Sep-2016.)
⊢ Ⅎ𝑥𝜑    &   ⊢ (𝜑 → Ⅎ𝑥𝜓)    &   ⊢ (𝜑 → (𝜓 → 𝜒))    ⇒   ⊢ (𝜑 → (𝜓 → ∀𝑥𝜒))
Theorem	alrimd 2253	Deduction form of Theorem 19.21 of [Margaris] p. 90, see 19.21 2245. (Contributed by Mario Carneiro, 24-Sep-2016.)
⊢ Ⅎ𝑥𝜑    &   ⊢ Ⅎ𝑥𝜓    &   ⊢ (𝜑 → (𝜓 → 𝜒))    ⇒   ⊢ (𝜑 → (𝜓 → ∀𝑥𝜒))
Theorem	eximd 2254	Deduction form of Theorem 19.22 of [Margaris] p. 90, see exim 1857. (Contributed by NM, 29-Jun-1993.) (Revised by Mario Carneiro, 24-Sep-2016.)
⊢ Ⅎ𝑥𝜑    &   ⊢ (𝜑 → (𝜓 → 𝜒))    ⇒   ⊢ (𝜑 → (∃𝑥𝜓 → ∃𝑥𝜒))
Theorem	exlimi 2255	Inference associated with 19.23 2249. See exlimiv 1953 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 2256	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 2257	Existential elimination rule of natural deduction. (Contributed by ML, 17-Jul-2020.) Shorten exlimdd 2258. (Revised by Wolf Lammen, 3-Sep-2023.)
⊢ Ⅎ𝑥𝜑    &   ⊢ Ⅎ𝑥𝜒    &   ⊢ (𝜑 → ∃𝑥𝜓)    &   ⊢ (𝜑 → (𝜓 → 𝜒))    ⇒   ⊢ (𝜑 → 𝜒)
Theorem	exlimdd 2258	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 2259	Deduction for generalization rule for negated wff. (Contributed by Mario Carneiro, 24-Sep-2016.)
⊢ Ⅎ𝑥𝜑    &   ⊢ (𝜑 → ¬ 𝜓)    ⇒   ⊢ (𝜑 → ¬ ∃𝑥𝜓)
Theorem	albid 2260	Formula-building rule for universal quantifier (deduction form). (Contributed by Mario Carneiro, 24-Sep-2016.)
⊢ Ⅎ𝑥𝜑    &   ⊢ (𝜑 → (𝜓 ↔ 𝜒))    ⇒   ⊢ (𝜑 → (∀𝑥𝜓 ↔ ∀𝑥𝜒))
Theorem	exbid 2261	Formula-building rule for existential quantifier (deduction form). (Contributed by Mario Carneiro, 24-Sep-2016.)
⊢ Ⅎ𝑥𝜑    &   ⊢ (𝜑 → (𝜓 ↔ 𝜒))    ⇒   ⊢ (𝜑 → (∃𝑥𝜓 ↔ ∃𝑥𝜒))
Theorem	nfbidf 2262	An equality theorem for effectively not free. (Contributed by Mario Carneiro, 4-Oct-2016.) df-nf 1807 changed. (Revised by Wolf Lammen, 18-Sep-2021.)
⊢ Ⅎ𝑥𝜑    &   ⊢ (𝜑 → (𝜓 ↔ 𝜒))    ⇒   ⊢ (𝜑 → (Ⅎ𝑥𝜓 ↔ Ⅎ𝑥𝜒))
Theorem	19.16 2263	Theorem 19.16 of [Margaris] p. 90. (Contributed by NM, 12-Mar-1993.)
⊢ Ⅎ𝑥𝜑    ⇒   ⊢ (∀𝑥(𝜑 ↔ 𝜓) → (𝜑 ↔ ∀𝑥𝜓))
Theorem	19.17 2264	Theorem 19.17 of [Margaris] p. 90. (Contributed by NM, 12-Mar-1993.)
⊢ Ⅎ𝑥𝜓    ⇒   ⊢ (∀𝑥(𝜑 ↔ 𝜓) → (∀𝑥𝜑 ↔ 𝜓))
Theorem	19.27 2265	Theorem 19.27 of [Margaris] p. 90. See 19.27v 2018 for a version requiring fewer axioms. (Contributed by NM, 21-Jun-1993.)
⊢ Ⅎ𝑥𝜓    ⇒   ⊢ (∀𝑥(𝜑 ∧ 𝜓) ↔ (∀𝑥𝜑 ∧ 𝜓))
Theorem	19.28 2266	Theorem 19.28 of [Margaris] p. 90. See 19.28v 2019 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 2267	Theorem 19.19 of [Margaris] p. 90. (Contributed by NM, 12-Mar-1993.)
⊢ Ⅎ𝑥𝜑    ⇒   ⊢ (∀𝑥(𝜑 ↔ 𝜓) → (𝜑 ↔ ∃𝑥𝜓))
Theorem	19.36 2268	Theorem 19.36 of [Margaris] p. 90. See 19.36v 2016 for a version requiring fewer axioms. (Contributed by NM, 24-Jun-1993.)
⊢ Ⅎ𝑥𝜓    ⇒   ⊢ (∃𝑥(𝜑 → 𝜓) ↔ (∀𝑥𝜑 → 𝜓))
Theorem	19.36i 2269	Inference associated with 19.36 2268. See 19.36iv 1969 for a version requiring fewer axioms. (Contributed by NM, 24-Jun-1993.)
⊢ Ⅎ𝑥𝜓    &   ⊢ ∃𝑥(𝜑 → 𝜓)    ⇒   ⊢ (∀𝑥𝜑 → 𝜓)
Theorem	19.37 2270	Theorem 19.37 of [Margaris] p. 90. See 19.37v 2020 for a version requiring fewer axioms. (Contributed by NM, 21-Jun-1993.)
⊢ Ⅎ𝑥𝜑    ⇒   ⊢ (∃𝑥(𝜑 → 𝜓) ↔ (𝜑 → ∃𝑥𝜓))
Theorem	19.32 2271	Theorem 19.32 of [Margaris] p. 90. See 19.32v 1963 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 2272	Theorem 19.31 of [Margaris] p. 90. See 19.31v 1964 for a version requiring fewer axioms. (Contributed by NM, 14-May-1993.)
⊢ Ⅎ𝑥𝜓    ⇒   ⊢ (∀𝑥(𝜑 ∨ 𝜓) ↔ (∀𝑥𝜑 ∨ 𝜓))
Theorem	19.41 2273	Theorem 19.41 of [Margaris] p. 90. See 19.41v 1972 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 2274	Theorem 19.42 of [Margaris] p. 90. See 19.42v 1976 for a version requiring fewer axioms. See exan 1885 for an immediate version. (Contributed by NM, 18-Aug-1993.)
⊢ Ⅎ𝑥𝜑    ⇒   ⊢ (∃𝑥(𝜑 ∧ 𝜓) ↔ (𝜑 ∧ ∃𝑥𝜓))
Theorem	19.44 2275	Theorem 19.44 of [Margaris] p. 90. See 19.44v 2021 for a version requiring fewer axioms. (Contributed by NM, 12-Mar-1993.)
⊢ Ⅎ𝑥𝜓    ⇒   ⊢ (∃𝑥(𝜑 ∨ 𝜓) ↔ (∃𝑥𝜑 ∨ 𝜓))
Theorem	19.45 2276	Theorem 19.45 of [Margaris] p. 90. See 19.45v 2022 for a version requiring fewer axioms. (Contributed by NM, 12-Mar-1993.)
⊢ Ⅎ𝑥𝜑    ⇒   ⊢ (∃𝑥(𝜑 ∨ 𝜓) ↔ (𝜑 ∨ ∃𝑥𝜓))
Theorem	spimfv 2277*	Specialization, using implicit substitution. Version of spim 2421 with a disjoint variable condition, which does not require ax-13 2406. See spimvw 2009 for a version with two disjoint variable conditions, requiring fewer axioms, and spimv 2424 for another variant. (Contributed by NM, 10-Jan-1993.) (Revised by BJ, 31-May-2019.)
⊢ Ⅎ𝑥𝜓    &   ⊢ (𝑥 = 𝑦 → (𝜑 → 𝜓))    ⇒   ⊢ (∀𝑥𝜑 → 𝜓)
Theorem	chvarfv 2278*	Implicit substitution of 𝑦 for 𝑥 into a theorem. Version of chvar 2429 with a disjoint variable condition, which does not require ax-13 2406. (Contributed by Raph Levien, 9-Jul-2003.) (Revised by BJ, 31-May-2019.)
⊢ Ⅎ𝑥𝜓    &   ⊢ (𝑥 = 𝑦 → (𝜑 ↔ 𝜓))    &   ⊢ 𝜑    ⇒   ⊢ 𝜓
Theorem	cbv3v2 2279*	Version of cbv3 2431 with two disjoint variable conditions, which does not require ax-11 2194 nor ax-13 2406. (Contributed by BJ, 24-Jun-2019.) (Proof shortened by Wolf Lammen, 30-Aug-2021.)
⊢ Ⅎ𝑥𝜓    &   ⊢ (𝑥 = 𝑦 → (𝜑 → 𝜓))    ⇒   ⊢ (∀𝑥𝜑 → ∀𝑦𝜓)
Theorem	sbalex 2280*	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 2094. That both sides of the biconditional express proper substitution is proved by sb5 2313 and sb6 2121. The implication "to the left" is equs4v 2023 and does not require ax-10 2178 nor ax-12 2215. It also holds without disjoint variable condition if we allow more axioms (see equs4 2450). Theorem 6.2 of [Quine] p. 40. Theorem equs5 2494 replaces the disjoint variable condition with a distinctor antecedent. Theorem equs45f 2493 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 2306 in place of equsex 2452 in order to remove dependency on ax-13 2406. (Revised by BJ, 20-Dec-2020.) Revise to remove dependency on df-sb 2094. (Revised by BJ, 21-Sep-2024.) (Proof shortened by SN, 14-Aug-2025.)
⊢ (∃𝑥(𝑥 = 𝑡 ∧ 𝜑) ↔ ∀𝑥(𝑥 = 𝑡 → 𝜑))
Theorem	sbalexOLD 2281*	Obsolete version of sbalex 2280 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 2282*	Version of sb4a 2514 with a disjoint variable condition, which does not require ax-13 2406. The distinctor antecedent from sb4b 2509 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 2283	Deduction substituting both sides of an implication. (Contributed by Wolf Lammen, 24-Nov-2022.) Revise df-sb 2094. (Revised by Steven Nguyen, 9-Jul-2023.)
⊢ Ⅎ𝑥𝜑    &   ⊢ (𝜑 → (𝜓 → 𝜒))    ⇒   ⊢ (𝜑 → ([𝑦 / 𝑥]𝜓 → [𝑦 / 𝑥]𝜒))
Theorem	sbbid 2284	Deduction substituting both sides of a biconditional. (Contributed by NM, 30-Jun-1993.) Remove dependency on ax-10 2178 and ax-13 2406. (Revised by Wolf Lammen, 24-Nov-2022.) Revise df-sb 2094. (Revised by Steven Nguyen, 11-Jul-2023.)
⊢ Ⅎ𝑥𝜑    &   ⊢ (𝜑 → (𝜓 ↔ 𝜒))    ⇒   ⊢ (𝜑 → ([𝑦 / 𝑥]𝜓 ↔ [𝑦 / 𝑥]𝜒))
Theorem	2sbbid 2285	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 2286	An equality theorem for substitution. (Contributed by NM, 16-May-1993.) Revise df-sb 2094. (Revised by BJ, 22-Dec-2020.)
⊢ (𝑥 = 𝑡 → (𝜑 → [𝑡 / 𝑥]𝜑))
Theorem	sbequ2 2287	An equality theorem for substitution. (Contributed by NM, 16-May-1993.) Revise df-sb 2094. (Revised by BJ, 22-Dec-2020.) (Proof shortened by Wolf Lammen, 3-Feb-2024.)
⊢ (𝑥 = 𝑡 → ([𝑡 / 𝑥]𝜑 → 𝜑))
Theorem	stdpc7 2288	One of the two equality axioms of standard predicate calculus, called substitutivity of equality. (The other one is stdpc6 2051.) 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 2289	An equality theorem for substitution. (Contributed by NM, 14-May-1993.)
⊢ (𝑥 = 𝑦 → (𝜑 ↔ [𝑦 / 𝑥]𝜑))
Theorem	sbequ12r 2290	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 2291*	Elimination of substitution. Also see sbel2x 2508. (Contributed by NM, 5-Aug-1993.) Avoid ax-13 2406. (Revised by Wolf Lammen, 6-Aug-2023.) Avoid ax-10 2178. (Revised by GG, 20-Aug-2023.)
⊢ (𝜑 ↔ ∃𝑥(𝑥 = 𝑦 ∧ [𝑥 / 𝑦]𝜑))
Theorem	sbequ12a 2292	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 2293	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 2294*	A composition law for substitution. Version of sbco 2541 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 2295*	Obsolete version of sbcov 2294 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 2296*	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 2297*	Reverting substitution yields the original expression. Based on fewer axioms than sbid2v 2543, 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 2298*	Generalization of axc16 2299. Use the latter when sufficient. This proof only requires, on top of { ax-1 6-- ax-7 2031 }, Theorem ax12v 2216. (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 2406, along an idea of BJ. (Revised by Wolf Lammen, 30-Nov-2019.) (Revised by BJ, 7-Jul-2021.) Shorten axc11rv 2303. (Revised by Wolf Lammen, 11-Oct-2021.)
⊢ (∀𝑥 𝑥 = 𝑦 → (𝜑 → ∀𝑧𝜑))
Theorem	axc16 2299*	Proof of older axiom ax-c16 39528. (Contributed by NM, 8-Nov-2006.) (Revised by NM, 22-Sep-2017.)
⊢ (∀𝑥 𝑥 = 𝑦 → (𝜑 → ∀𝑥𝜑))
Theorem	axc16gb 2300*	Biconditional strengthening of axc16g 2298. (Contributed by NM, 15-May-1993.)
⊢ (∀𝑥 𝑥 = 𝑦 → (𝜑 ↔ ∀𝑧𝜑))