P. 22 of Theorem List - Metamath Proof Explorer

Theorem List for Metamath Proof Explorer - 2101-2200   *Has distinct variable group(s)

Type	Label	Description
Statement
Theorem	stdpc4 2101	The specialization axiom of standard predicate calculus. It states that if a statement 𝜑 holds for all 𝑥, then it also holds for the specific case of 𝑡 (properly) substituted for 𝑥. Translated to traditional notation, it can be read: "∀𝑥𝜑(𝑥) → 𝜑(𝑡), provided that 𝑡 is free for 𝑥 in 𝜑(𝑥)". Axiom 4 of [Mendelson] p. 69. See also spsbc 3760 and rspsbc 3835. (Contributed by NM, 14-May-1993.) Revise df-sb 2094. (Revised by BJ, 22-Dec-2020.) Revise df-sb again. (Revised by Wolf Lammen, 4-Jun-2026.)
⊢ (∀𝑥𝜑 → [𝑡 / 𝑥]𝜑)
Theorem	stdpc4ALT 2102	Alternate proof of stdpc4 2101, shorter but using additional axioms. (Contributed by WL, 5-Jun-2026.) (Proof modification is discouraged.) (New usage is discouraged.)
⊢ (∀𝑥𝜑 → [𝑡 / 𝑥]𝜑)
Theorem	sbtALT 2103	Alternate proof of sbt 2098, shorter but using additional axioms. (Contributed by NM, 21-Jan-2004.) (Proof modification is discouraged.) (New usage is discouraged.)
⊢ 𝜑    ⇒   ⊢ [𝑦 / 𝑥]𝜑
Theorem	2stdpc4 2104	A double specialization using explicit substitution. This is Theorem PM*11.1 in [WhiteheadRussell] p. 159. See stdpc4 2101 for the analogous single specialization. See 2sp 2224 for another double specialization. (Contributed by Andrew Salmon, 24-May-2011.)
⊢ (∀𝑥∀𝑦𝜑 → [𝑧 / 𝑥][𝑤 / 𝑦]𝜑)
Theorem	sbi1lem 2105*	Lemma for sbi1 2106. The core of the proof was extracted from a proof of SN. (Contributed by Wolf Lammen, 5-Jun-2026.)
⊢ (([𝑡 / 𝑥](𝜑 → 𝜓) ∧ [𝑡 / 𝑥]𝜑) → ∀𝑦(𝑦 = 𝑡 → ∀𝑥(𝑥 = 𝑦 → 𝜓)))
Theorem	sbi1 2106	Distribute substitution over implication. (Contributed by NM, 14-May-1993.) Remove dependencies on axioms. (Revised by Steven Nguyen, 24-Jul-2023.) Definition df-sb 2094 changed. (Revised by Wolf Lammen, 5-Jun-2026.)
⊢ ([𝑦 / 𝑥](𝜑 → 𝜓) → ([𝑦 / 𝑥]𝜑 → [𝑦 / 𝑥]𝜓))
Theorem	sbi1ALT 2107	Alternate proof of sbt 2098, shorter but using additional axioms. (Contributed by NM, 14-May-1993.) Remove dependencies on axioms. (Revised by Steven Nguyen, 24-Jul-2023.) (New usage is discouraged.) (Proof modification is discouraged.)
⊢ ([𝑦 / 𝑥](𝜑 → 𝜓) → ([𝑦 / 𝑥]𝜑 → [𝑦 / 𝑥]𝜓))
Theorem	spsbim 2108	Distribute substitution over implication. Closed form of sbimi 2110. Specialization of implication. (Contributed by NM, 5-Aug-1993.) (Proof shortened by Andrew Salmon, 25-May-2011.) Revise df-sb 2094. (Revised by BJ, 22-Dec-2020.) (Proof shortened by Steven Nguyen, 24-Jul-2023.)
⊢ (∀𝑥(𝜑 → 𝜓) → ([𝑡 / 𝑥]𝜑 → [𝑡 / 𝑥]𝜓))
Theorem	spsbbi 2109	Biconditional property for substitution. Closed form of sbbii 2112. Specialization of biconditional. (Contributed by NM, 2-Jun-1993.) Revise df-sb 2094. (Revised by BJ, 22-Dec-2020.)
⊢ (∀𝑥(𝜑 ↔ 𝜓) → ([𝑡 / 𝑥]𝜑 ↔ [𝑡 / 𝑥]𝜓))
Theorem	sbimi 2110	Distribute substitution over implication. (Contributed by NM, 25-Jun-1998.) Revise df-sb 2094. (Revised by BJ, 22-Dec-2020.) (Proof shortened by Steven Nguyen, 24-Jul-2023.)
⊢ (𝜑 → 𝜓)    ⇒   ⊢ ([𝑡 / 𝑥]𝜑 → [𝑡 / 𝑥]𝜓)
Theorem	sb2imi 2111	Distribute substitution over implication. Compare al2imi 1838. (Contributed by Steven Nguyen, 13-Aug-2023.)
⊢ (𝜑 → (𝜓 → 𝜒))    ⇒   ⊢ ([𝑡 / 𝑥]𝜑 → ([𝑡 / 𝑥]𝜓 → [𝑡 / 𝑥]𝜒))
Theorem	sbbii 2112	Infer substitution into both sides of a logical equivalence. (Contributed by NM, 14-May-1993.)
⊢ (𝜑 ↔ 𝜓)    ⇒   ⊢ ([𝑡 / 𝑥]𝜑 ↔ [𝑡 / 𝑥]𝜓)
Theorem	2sbbii 2113	Infer double substitution into both sides of a logical equivalence. (Contributed by AV, 30-Jul-2023.)
⊢ (𝜑 ↔ 𝜓)    ⇒   ⊢ ([𝑡 / 𝑥][𝑢 / 𝑦]𝜑 ↔ [𝑡 / 𝑥][𝑢 / 𝑦]𝜓)
Theorem	sbimdv 2114*	Deduction substituting both sides of an implication, with 𝜑 and 𝑥 disjoint. See also sbimd 2283. (Contributed by Wolf Lammen, 6-May-2023.) Revise df-sb 2094. (Revised by Steven Nguyen, 6-Jul-2023.)
⊢ (𝜑 → (𝜓 → 𝜒))    ⇒   ⊢ (𝜑 → ([𝑡 / 𝑥]𝜓 → [𝑡 / 𝑥]𝜒))
Theorem	sbbidv 2115*	Deduction substituting both sides of a biconditional, with 𝜑 and 𝑥 disjoint. See also sbbid 2284. (Contributed by Wolf Lammen, 6-May-2023.) (Proof shortened by Steven Nguyen, 6-Jul-2023.)
⊢ (𝜑 → (𝜓 ↔ 𝜒))    ⇒   ⊢ (𝜑 → ([𝑡 / 𝑥]𝜓 ↔ [𝑡 / 𝑥]𝜒))
Theorem	sban 2116	Conjunction inside and outside of a substitution are equivalent. Compare 19.26 1893. (Contributed by NM, 14-May-1993.) (Proof shortened by Steven Nguyen, 13-Aug-2023.)
⊢ ([𝑦 / 𝑥](𝜑 ∧ 𝜓) ↔ ([𝑦 / 𝑥]𝜑 ∧ [𝑦 / 𝑥]𝜓))
Theorem	sb3an 2117	Threefold conjunction inside and outside of a substitution are equivalent. (Contributed by NM, 14-Dec-2006.)
⊢ ([𝑦 / 𝑥](𝜑 ∧ 𝜓 ∧ 𝜒) ↔ ([𝑦 / 𝑥]𝜑 ∧ [𝑦 / 𝑥]𝜓 ∧ [𝑦 / 𝑥]𝜒))
Theorem	spsbe 2118	Existential generalization: if a proposition is true for a specific instance, then there exists an instance where it is true. (Contributed by NM, 29-Jun-1993.) (Proof shortened by Wolf Lammen, 3-May-2018.) Revise df-sb 2094. (Revised by BJ, 22-Dec-2020.) (Proof shortened by Steven Nguyen, 11-Jul-2023.) Revise df-sb 2094. (Revised by Wolf Lammen, 4-Jun-2026.)
⊢ ([𝑡 / 𝑥]𝜑 → ∃𝑥𝜑)
Theorem	sbequ 2119	Equality property for substitution, from Tarski's system. Used in proof of Theorem 9.7 in [Megill] p. 449 (p. 16 of the preprint). (Contributed by NM, 14-May-1993.) Revise df-sb 2094. (Revised by BJ, 30-Dec-2020.)
⊢ (𝑥 = 𝑦 → ([𝑥 / 𝑧]𝜑 ↔ [𝑦 / 𝑧]𝜑))
Theorem	sbequi 2120	An equality theorem for substitution. (Contributed by NM, 14-May-1993.) (Proof shortened by Wolf Lammen, 15-Sep-2018.) (Proof shortened by Steven Nguyen, 7-Jul-2023.)
⊢ (𝑥 = 𝑦 → ([𝑥 / 𝑧]𝜑 → [𝑦 / 𝑧]𝜑))
Theorem	sb6 2121*	Alternate definition of substitution when variables are disjoint. Compare Theorem 6.2 of [Quine] p. 40. Also proved as Lemmas 16 and 17 of [Tarski] p. 70. The implication "to the left" also holds without a disjoint variable condition (sb2 2513). Theorem sb6f 2531 replaces the disjoint variable condition with a nonfreeness hypothesis. Theorem sb4b 2509 replaces it with a distinctor antecedent. (Contributed by NM, 18-Aug-1993.) (Proof shortened by Wolf Lammen, 21-Sep-2018.) Revise df-sb 2094. (Revised by BJ, 22-Dec-2020.) Remove use of ax-11 2194. (Revised by Steven Nguyen, 7-Jul-2023.) (Proof shortened by Wolf Lammen, 16-Jul-2023.)
⊢ ([𝑡 / 𝑥]𝜑 ↔ ∀𝑥(𝑥 = 𝑡 → 𝜑))
Theorem	2sb6 2122*	Equivalence for double substitution. (Contributed by NM, 3-Feb-2005.)
⊢ ([𝑧 / 𝑥][𝑤 / 𝑦]𝜑 ↔ ∀𝑥∀𝑦((𝑥 = 𝑧 ∧ 𝑦 = 𝑤) → 𝜑))
Theorem	sb1v 2123*	One direction of sb5 2313, provable from fewer axioms. Version of sb1 2512 with a disjoint variable condition using fewer axioms. (Contributed by NM, 13-May-1993.) (Revised by Wolf Lammen, 20-Jan-2024.)
⊢ ([𝑦 / 𝑥]𝜑 → ∃𝑥(𝑥 = 𝑦 ∧ 𝜑))
Theorem	sbv 2124*	Substitution for a variable not occurring in a proposition. See sbf 2308 for a version without disjoint variable condition on 𝑥, 𝜑. If one adds a disjoint variable condition on 𝑥, 𝑡, then sbv 2124 can be proved directly by chaining equsv 2026 with sb6 2121. (Contributed by BJ, 22-Dec-2020.)
⊢ ([𝑡 / 𝑥]𝜑 ↔ 𝜑)
Theorem	sbcom4 2125*	Commutativity law for substitution. This theorem was incorrectly used as our previous version of pm11.07 2126 but may still be useful. (Contributed by Andrew Salmon, 17-Jun-2011.) (Proof shortened by Jim Kingdon, 22-Jan-2018.)
⊢ ([𝑤 / 𝑥][𝑦 / 𝑧]𝜑 ↔ [𝑦 / 𝑥][𝑤 / 𝑧]𝜑)
Theorem	pm11.07 2126	Axiom *11.07 in [WhiteheadRussell] p. 159. The original reads: *11.07 "Whatever possible argument 𝑥 may be, 𝜑(𝑥, 𝑦) is true whatever possible argument 𝑦 may be" implies the corresponding statement with 𝑥 and 𝑦 interchanged except in "𝜑(𝑥, 𝑦)". Under our formalism this appears to correspond to idi 1 and not to sbcom4 2125 as earlier thought. See https://groups.google.com/g/metamath/c/iS0fOvSemC8/m/M1zTH8wxCAAJ 2125. (Contributed by BJ, 16-Sep-2018.) (New usage is discouraged.)
⊢ 𝜑    ⇒   ⊢ 𝜑
Theorem	sbrimvw 2127*	Substitution in an implication with a variable not free in the antecedent affects only the consequent. Version of sbrim 2341 based on fewer axioms, but with more disjoint variable conditions. (Contributed by Wolf Lammen, 29-Jan-2024.) Remove DV condition. (Revised by Wolf Lammen, 5-Jun-2026.)
⊢ ([𝑦 / 𝑥](𝜑 → 𝜓) ↔ (𝜑 → [𝑦 / 𝑥]𝜓))
Theorem	sbrimvwOLD 2128*	Obsolete version of sbrimvw 2127 as of 5-Jun-2026. (Contributed by Wolf Lammen, 29-Jan-2024.) (Proof modification is discouraged.) (New usage is discouraged.)
⊢ ([𝑦 / 𝑥](𝜑 → 𝜓) ↔ (𝜑 → [𝑦 / 𝑥]𝜓))
Theorem	sbbiiev 2129*	An equivalence of substitutions (as in sbbii 2112) allowing the additional information that 𝑥 = 𝑡. Version of sbiev 2349 and sbievw 2130 without a disjoint variable condition on 𝜓, useful for substituting only part of 𝜑. (Contributed by SN, 24-Aug-2025.)
⊢ (𝑥 = 𝑡 → (𝜑 ↔ 𝜓))    ⇒   ⊢ ([𝑡 / 𝑥]𝜑 ↔ [𝑡 / 𝑥]𝜓)
Theorem	sbievw 2130*	Conversion of implicit substitution to explicit substitution. Version of sbie 2536 and sbiev 2349 with more disjoint variable conditions, requiring fewer axioms. (Contributed by NM, 30-Jun-1994.) (Revised by BJ, 18-Jul-2023.) (Proof shortened by SN, 24-Aug-2025.)
⊢ (𝑥 = 𝑦 → (𝜑 ↔ 𝜓))    ⇒   ⊢ ([𝑦 / 𝑥]𝜑 ↔ 𝜓)
Theorem	sbievwOLD 2131*	Obsolete version of sbievw 2130 as of 24-Aug-2025. (Contributed by NM, 30-Jun-1994.) (Revised by BJ, 18-Jul-2023.) (Proof modification is discouraged.) (New usage is discouraged.)
⊢ (𝑥 = 𝑦 → (𝜑 ↔ 𝜓))    ⇒   ⊢ ([𝑦 / 𝑥]𝜑 ↔ 𝜓)
Theorem	sbiedvw 2132*	Conversion of implicit substitution to explicit substitution (deduction version of sbievw 2130). Version of sbied 2537 and sbiedv 2538 with more disjoint variable conditions, requiring fewer axioms. (Contributed by NM, 30-Jun-1994.) (Revised by GG, 29-Jan-2024.)
⊢ ((𝜑 ∧ 𝑥 = 𝑦) → (𝜓 ↔ 𝜒))    ⇒   ⊢ (𝜑 → ([𝑦 / 𝑥]𝜓 ↔ 𝜒))
Theorem	2sbievw 2133*	Conversion of double implicit substitution to explicit substitution. Version of 2sbiev 2539 with more disjoint variable conditions, requiring fewer axioms. (Contributed by AV, 29-Jul-2023.) Avoid ax-13 2406. (Revised by GG, 10-Jan-2024.)
⊢ ((𝑥 = 𝑡 ∧ 𝑦 = 𝑢) → (𝜑 ↔ 𝜓))    ⇒   ⊢ ([𝑡 / 𝑥][𝑢 / 𝑦]𝜑 ↔ 𝜓)
Theorem	sbcom3vv 2134*	Substituting 𝑦 for 𝑥 and then 𝑧 for 𝑦 is equivalent to substituting 𝑧 for both 𝑥 and 𝑦. Version of sbcom3 2540 with a disjoint variable condition using fewer axioms. (Contributed by NM, 27-May-1997.) (Revised by Giovanni Mascellani, 8-Apr-2018.) (Revised by BJ, 30-Dec-2020.) (Proof shortened by Wolf Lammen, 19-Jan-2023.)
⊢ ([𝑧 / 𝑦][𝑦 / 𝑥]𝜑 ↔ [𝑧 / 𝑦][𝑧 / 𝑥]𝜑)
Theorem	sbievw2 2135*	sbievw 2130 applied twice, avoiding a DV condition on 𝑥, 𝑦. Based on proofs by Wolf Lammen. (Contributed by Steven Nguyen, 29-Jul-2023.)
⊢ (𝑥 = 𝑤 → (𝜑 ↔ 𝜒))    &   ⊢ (𝑤 = 𝑦 → (𝜒 ↔ 𝜓))    ⇒   ⊢ ([𝑦 / 𝑥]𝜑 ↔ 𝜓)
Theorem	sbco2vv 2136*	A composition law for substitution. Version of sbco2 2545 with disjoint variable conditions and fewer axioms. (Contributed by NM, 30-Jun-1994.) (Revised by BJ, 22-Dec-2020.) (Proof shortened by Wolf Lammen, 29-Apr-2023.)
⊢ ([𝑦 / 𝑧][𝑧 / 𝑥]𝜑 ↔ [𝑦 / 𝑥]𝜑)
Theorem	cbvsbv 2137*	Change the bound variable (i.e. the substituted one) in wff's linked by implicit substitution. The proof was extracted from a former cbvabv 2835 version. (Contributed by Wolf Lammen, 16-Mar-2025.)
⊢ (𝑥 = 𝑦 → (𝜑 ↔ 𝜓))    ⇒   ⊢ ([𝑧 / 𝑥]𝜑 ↔ [𝑧 / 𝑦]𝜓)
Theorem	sbco4lem 2138*	Lemma for sbco4 2139. It replaces the temporary variable 𝑣 with another temporary variable 𝑤. (Contributed by Jim Kingdon, 26-Sep-2018.) (Proof shortened by Wolf Lammen, 12-Oct-2024.) Avoid ax-11 2194. (Revised by SN, 3-Sep-2025.)
⊢ ([𝑥 / 𝑣][𝑦 / 𝑥][𝑣 / 𝑦]𝜑 ↔ [𝑥 / 𝑤][𝑦 / 𝑥][𝑤 / 𝑦]𝜑)
Theorem	sbco4 2139*	Two ways of exchanging two variables. Both sides of the biconditional exchange 𝑥 and 𝑦, either via two temporary variables 𝑢 and 𝑣, or a single temporary 𝑤. (Contributed by Jim Kingdon, 25-Sep-2018.) Avoid ax-11 2194. (Revised by SN, 3-Sep-2025.)
⊢ ([𝑦 / 𝑢][𝑥 / 𝑣][𝑢 / 𝑥][𝑣 / 𝑦]𝜑 ↔ [𝑥 / 𝑤][𝑦 / 𝑥][𝑤 / 𝑦]𝜑)
Theorem	equsb3 2140*	Substitution in an equality. (Contributed by Raph Levien and FL, 4-Dec-2005.) Reduce axiom usage. (Revised by Wolf Lammen, 23-Jul-2023.)
⊢ ([𝑦 / 𝑥]𝑥 = 𝑧 ↔ 𝑦 = 𝑧)
Theorem	equsb3r 2141*	Substitution applied to the atomic wff with equality. Variant of equsb3 2140. (Contributed by AV, 29-Jul-2023.) (Proof shortened by Wolf Lammen, 2-Sep-2023.)
⊢ ([𝑦 / 𝑥]𝑧 = 𝑥 ↔ 𝑧 = 𝑦)
Theorem	equsb1v 2142*	Substitution applied to an atomic wff. Version of equsb1 2525 with a disjoint variable condition, which neither requires ax-12 2215 nor ax-13 2406. (Contributed by NM, 10-May-1993.) (Revised by BJ, 11-Sep-2019.) Remove dependencies on axioms. (Revised by Wolf Lammen, 30-May-2023.) (Proof shortened by Steven Nguyen, 19-Jun-2023.) Revise df-sb 2094. (Revised by Steven Nguyen, 11-Jul-2023.) (Proof shortened by Steven Nguyen, 22-Jul-2023.)
⊢ [𝑦 / 𝑥]𝑥 = 𝑦
Theorem	nsb 2143	Any substitution in an always false formula is false. (Contributed by Steven Nguyen, 3-May-2023.)
⊢ (∀𝑥 ¬ 𝜑 → ¬ [𝑡 / 𝑥]𝜑)
Theorem	sbn1 2144	One direction of sbn 2317, using fewer axioms. Compare 19.2 1999. (Contributed by Steven Nguyen, 18-Aug-2023.)
⊢ ([𝑡 / 𝑥] ¬ 𝜑 → ¬ [𝑡 / 𝑥]𝜑)
1.4.9  Membership predicate
Syntax	wcel 2145	Extend wff definition to include the membership connective between classes. For a general discussion of the theory of classes, see mmset.html#class. The purpose of introducing wff 𝐴 ∈ 𝐵 here is to allow to prove the wel 2146 of predicate calculus in terms of the wcel 2145 of set theory, so that we do not overload the ∈ connective with two syntax definitions. This is done to prevent ambiguity that would complicate some Metamath parsers. The class variables 𝐴 and 𝐵 are introduced temporarily for the purpose of this definition but otherwise not used in predicate calculus. See df-clab 2744 for more information on the set theory usage of wcel 2145.
wff 𝐴 ∈ 𝐵
Theorem	wel 2146	Extend wff definition to include atomic formulas with the membership predicate. This is read either "𝑥 is an element of 𝑦", or "𝑥 is a member of 𝑦", or "𝑥 belongs to 𝑦", or "𝑦 contains 𝑥". Note: The phrase "𝑦 includes 𝑥 " means "𝑥 is a subset of 𝑦"; to use it also for 𝑥 ∈ 𝑦, as some authors occasionally do, is poor form and causes confusion, according to George Boolos (1992 lecture at MIT). This syntactic construction introduces a binary non-logical predicate symbol ∈ (stylized lowercase epsilon) into our predicate calculus. We will eventually use it for the membership predicate of set theory, but that is irrelevant at this point: the predicate calculus axioms for ∈ apply to any arbitrary binary predicate symbol. "Non-logical" means that the predicate is presumed to have additional properties beyond the realm of predicate calculus, although these additional properties are not specified by predicate calculus itself but rather by the axioms of a theory (in our case set theory) added to predicate calculus. "Binary" means that the predicate has two arguments. Instead of introducing wel 2146 as an axiomatic statement, as was done in an older version of this database, we introduce it by "proving" a special case of set theory's more general wcel 2145. This lets us avoid overloading the ∈ connective, thus preventing ambiguity that would complicate certain Metamath parsers. However, logically wel 2146 is considered to be a primitive syntax, even though here it is artificially "derived" from wcel 2145. Note: To see the proof steps of this syntax proof, type "MM> SHOW PROOF wel / ALL" in the Metamath program. (Contributed by NM, 24-Jan-2006.)
wff 𝑥 ∈ 𝑦
1.4.10  Axiom scheme ax-8 (Left Equality for Binary Predicate)
Axiom	ax-8 2147	Axiom of Left Equality for Binary Predicate. One of the equality and substitution axioms for a non-logical predicate in our predicate calculus with equality. It substitutes equal variables into the left-hand side of an arbitrary binary predicate ∈, which we will use for the set membership relation when set theory is introduced. This axiom scheme is a sub-scheme of Axiom Scheme B8 of system S2 of [Tarski], p. 75, whose general form cannot be represented with our notation. Also appears as Axiom scheme C12' in [Megill] p. 448 (p. 16 of the preprint). "Non-logical" means that the predicate is not a primitive of predicate calculus proper but instead is an extension to it. "Binary" means that the predicate has two arguments. In a system of predicate calculus with equality, like ours, equality is not usually considered to be a non-logical predicate. In systems of predicate calculus without equality, it typically would be. We prove in ax8 2151 that this axiom can be recovered from its weakened version ax8v 2148 where 𝑥 and 𝑦 are assumed to be disjoint variables. In particular, the only theorem referencing ax-8 2147 should be ax8v 2148. See the comment of ax8v 2148 for more details on these matters. (Contributed by NM, 30-Jun-1993.) (Revised by BJ, 7-Dec-2020.) Use ax8 2151 instead. (New usage is discouraged.)
⊢ (𝑥 = 𝑦 → (𝑥 ∈ 𝑧 → 𝑦 ∈ 𝑧))
Theorem	ax8v 2148*	Weakened version of ax-8 2147, with a disjoint variable condition on 𝑥, 𝑦. This should be the only proof referencing ax-8 2147, and it should be referenced only by its two weakened versions ax8v1 2149 and ax8v2 2150, from which ax-8 2147 is then rederived as ax8 2151, which shows that either ax8v 2148 or the conjunction of ax8v1 2149 and ax8v2 2150 is sufficient. (Contributed by BJ, 7-Dec-2020.) Use ax8 2151 instead. (New usage is discouraged.)
⊢ (𝑥 = 𝑦 → (𝑥 ∈ 𝑧 → 𝑦 ∈ 𝑧))
Theorem	ax8v1 2149*	First of two weakened versions of ax8v 2148, with an extra disjoint variable condition on 𝑥, 𝑧, see comments there. (Contributed by BJ, 7-Dec-2020.)
⊢ (𝑥 = 𝑦 → (𝑥 ∈ 𝑧 → 𝑦 ∈ 𝑧))
Theorem	ax8v2 2150*	Second of two weakened versions of ax8v 2148, with an extra disjoint variable condition on 𝑦, 𝑧 see comments there. (Contributed by BJ, 7-Dec-2020.)
⊢ (𝑥 = 𝑦 → (𝑥 ∈ 𝑧 → 𝑦 ∈ 𝑧))
Theorem	ax8 2151	Proof of ax-8 2147 from ax8v1 2149 and ax8v2 2150, proving sufficiency of the conjunction of the latter two weakened versions of ax8v 2148, which is itself a weakened version of ax-8 2147. (Contributed by BJ, 7-Dec-2020.) (Proof shortened by Wolf Lammen, 11-Apr-2021.)
⊢ (𝑥 = 𝑦 → (𝑥 ∈ 𝑧 → 𝑦 ∈ 𝑧))
Theorem	elequ1 2152	An identity law for the non-logical predicate. (Contributed by NM, 30-Jun-1993.)
⊢ (𝑥 = 𝑦 → (𝑥 ∈ 𝑧 ↔ 𝑦 ∈ 𝑧))
Theorem	elsb1 2153*	Substitution for the first argument of the non-logical predicate in an atomic formula. See elsb2 2162 for substitution for the second argument. (Contributed by NM, 7-Nov-2006.) (Proof shortened by Andrew Salmon, 14-Jun-2011.) Reduce axiom usage. (Revised by Wolf Lammen, 24-Jul-2023.)
⊢ ([𝑦 / 𝑥]𝑥 ∈ 𝑧 ↔ 𝑦 ∈ 𝑧)
Theorem	cleljust 2154*	When the class variables in Definition df-clel 2840 are replaced with setvar variables, this theorem of predicate calculus is the result. This theorem provides part of the justification for the consistency of that definition, which "overloads" the setvar variables in wel 2146 with the class variables in wcel 2145. (Contributed by NM, 28-Jan-2004.) Revised to use equsexvw 2028 in order to remove dependencies on ax-10 2178, ax-12 2215, ax-13 2406. Note that there is no disjoint variable condition on 𝑥, 𝑦, that is, on the variables of the left-hand side, as should be the case for definitions. (Revised by BJ, 29-Dec-2020.)
⊢ (𝑥 ∈ 𝑦 ↔ ∃𝑧(𝑧 = 𝑥 ∧ 𝑧 ∈ 𝑦))
1.4.11  Axiom scheme ax-9 (Right Equality for Binary Predicate)
Axiom	ax-9 2155	Axiom of Right Equality for Binary Predicate. One of the equality and substitution axioms for a non-logical predicate in our predicate calculus with equality. It substitutes equal variables into the right-hand side of an arbitrary binary predicate ∈, which we will use for the set membership relation when set theory is introduced. This axiom scheme is a sub-scheme of Axiom Scheme B8 of system S2 of [Tarski], p. 75, whose general form cannot be represented with our notation. Also appears as Axiom scheme C13' in [Megill] p. 448 (p. 16 of the preprint). We prove in ax9 2159 that this axiom can be recovered from its weakened version ax9v 2156 where 𝑥 and 𝑦 are assumed to be disjoint variables. In particular, the only theorem referencing ax-9 2155 should be ax9v 2156. See the comment of ax9v 2156 for more details on these matters. (Contributed by NM, 21-Jun-1993.) (Revised by BJ, 7-Dec-2020.) Use ax9 2159 instead. (New usage is discouraged.)
⊢ (𝑥 = 𝑦 → (𝑧 ∈ 𝑥 → 𝑧 ∈ 𝑦))
Theorem	ax9v 2156*	Weakened version of ax-9 2155, with a disjoint variable condition on 𝑥, 𝑦. This should be the only proof referencing ax-9 2155, and it should be referenced only by its two weakened versions ax9v1 2157 and ax9v2 2158, from which ax-9 2155 is then rederived as ax9 2159, which shows that either ax9v 2156 or the conjunction of ax9v1 2157 and ax9v2 2158 is sufficient. (Contributed by BJ, 7-Dec-2020.) Use ax9 2159 instead. (New usage is discouraged.)
⊢ (𝑥 = 𝑦 → (𝑧 ∈ 𝑥 → 𝑧 ∈ 𝑦))
Theorem	ax9v1 2157*	First of two weakened versions of ax9v 2156, with an extra disjoint variable condition on 𝑥, 𝑧, see comments there. (Contributed by BJ, 7-Dec-2020.)
⊢ (𝑥 = 𝑦 → (𝑧 ∈ 𝑥 → 𝑧 ∈ 𝑦))
Theorem	ax9v2 2158*	Second of two weakened versions of ax9v 2156, with an extra disjoint variable condition on 𝑦, 𝑧 see comments there. (Contributed by BJ, 7-Dec-2020.)
⊢ (𝑥 = 𝑦 → (𝑧 ∈ 𝑥 → 𝑧 ∈ 𝑦))
Theorem	ax9 2159	Proof of ax-9 2155 from ax9v1 2157 and ax9v2 2158, proving sufficiency of the conjunction of the latter two weakened versions of ax9v 2156, which is itself a weakened version of ax-9 2155. (Contributed by BJ, 7-Dec-2020.) (Proof shortened by Wolf Lammen, 11-Apr-2021.)
⊢ (𝑥 = 𝑦 → (𝑧 ∈ 𝑥 → 𝑧 ∈ 𝑦))
Theorem	elequ2 2160	An identity law for the non-logical predicate. (Contributed by NM, 21-Jun-1993.)
⊢ (𝑥 = 𝑦 → (𝑧 ∈ 𝑥 ↔ 𝑧 ∈ 𝑦))
Theorem	elequ2g 2161*	A form of elequ2 2160 with a universal quantifier. Its converse is the axiom of extensionality ax-ext 2737. (Contributed by BJ, 3-Oct-2019.)
⊢ (𝑥 = 𝑦 → ∀𝑧(𝑧 ∈ 𝑥 ↔ 𝑧 ∈ 𝑦))
Theorem	elsb2 2162*	Substitution for the second argument of the non-logical predicate in an atomic formula. See elsb1 2153 for substitution for the first argument. (Contributed by Rodolfo Medina, 3-Apr-2010.) (Proof shortened by Andrew Salmon, 14-Jun-2011.) Reduce axiom usage. (Revised by Wolf Lammen, 24-Jul-2023.)
⊢ ([𝑦 / 𝑥]𝑧 ∈ 𝑥 ↔ 𝑧 ∈ 𝑦)
Theorem	elequ12 2163	An identity law for the non-logical predicate, which combines elequ1 2152 and elequ2 2160. The analogous theorems for class terms are eleq1 2853, eleq2 2854, and eleq12 2855 respectively. (Contributed by BJ, 29-Sep-2019.)
⊢ ((𝑥 = 𝑦 ∧ 𝑧 = 𝑡) → (𝑥 ∈ 𝑧 ↔ 𝑦 ∈ 𝑡))
Theorem	ru0 2164*	The FOL statement used in the standard proof of Russell's paradox ru 3746. (Contributed by NM, 7-Aug-1994.) Extract from proof of ru 3746 and reduce axiom usage. (Revised by BJ, 12-Oct-2019.)
⊢ ¬ ∀𝑥(𝑥 ∈ 𝑦 ↔ ¬ 𝑥 ∈ 𝑥)
1.4.12  Logical redundancy of ax-10 , ax-11 , ax-12 , ax-13 The original axiom schemes of Tarski's predicate calculus are ax-4 1832, ax-5 1933, ax6v 1991, ax-7 2031, ax-8 2147, and ax-9 2155, together with rule ax-gen 1818. See mmset.html#compare 1818. They are given as axiom schemes B4 through B8 in [KalishMontague] p. 81. These are shown to be logically complete by Theorem 1 of [KalishMontague] p. 85. The axiom system of set.mm includes the auxiliary axiom schemes ax-10 2178, ax-11 2194, ax-12 2215, and ax-13 2406, which are not part of Tarski's axiom schemes. Each object-language instance of them is provable from Tarski's axioms, so they are logically redundant. However, they are conjectured not to be provable directly as schemes from Tarski's axiom schemes using only Metamath's direct substitution rule. They are used to make our system "scheme complete", i.e., able to prove directly all possible schemes with wff and setvar variables, bundled or not, whose object-language instances are valid. (ax-12 2215 has been proved to be required; see https://us.metamath.org/award2003.html#9a 2215. Metalogical independence of the other three are open problems.) (There are additional predicate calculus axiom schemes included in set.mm such as ax-c5 39519, but they can all be proved as theorems from the above.) Terminology: Two setvar (individual) metavariables are "bundled" in an axiom or theorem scheme when there is no distinct variable constraint ($d) imposed on them. (The term "bundled" is due to Raph Levien.) For example, the 𝑥 and 𝑦 in ax-6 1990 are bundled, but they are not in ax6v 1991. We also say that a scheme is bundled when it has at least one pair of bundled setvar variables. If distinct variable conditions are added to all setvar variable pairs in a bundled scheme, we call that the "principal" instance of the bundled scheme. For example, ax6v 1991 is the principal instance of ax-6 1990. Whenever a common variable is substituted for two or more bundled variables in an axiom or theorem scheme, we call the substitution instance "degenerate". For example, the instance ¬ ∀𝑥¬ 𝑥 = 𝑥 of ax-6 1990 is degenerate. An advantage of bundling is ease of use since there are fewer distinct variable restrictions ($d) to be concerned with, and theorems are more general. There may be some economy in being able to prove facts about principal and degenerate instances simultaneously. A disadvantage is that bundling may present difficulties in translations to other proof languages, which typically lack the concept (in part because their variables often represent the variables of the object language rather than metavariables ranging over them). Because Tarski's axiom schemes are logically complete, they can be used to prove any object-language instance of ax-10 2178, ax-11 2194, ax-12 2215, and ax-13 2406. "Translating" this to Metamath, it means that Tarski's axioms can prove any substitution instance of ax-10 2178, ax-11 2194, ax-12 2215, or ax-13 2406 in which (1) there are no wff metavariables and (2) all setvar variables are mutually distinct i.e. are not bundled. In effect this is mimicking the object language by pretending that each setvar variable is an object-language variable. (There may also be specific instances with wff metavariables and/or bundling that are directly provable from Tarski's axiom schemes, but it isn't guaranteed. Whether all of them are possible is part of the still open metalogical independence problem for our additional axiom schemes.) It can be useful to see how this can be done, both to show that our additional schemes are valid metatheorems of Tarski's system and to be able to translate object-language instances of our proofs into proofs that would work with a system using only Tarski's original schemes. In addition, it may (or may not) provide insight into the conjectured metalogical independence of our additional schemes. The theorem schemes ax10w 2166, ax11w 2167, ax12w 2170, and ax13w 2173 are derived using only Tarski's axiom schemes, showing that Tarski's schemes can be used to derive all substitution instances of ax-10 2178, ax-11 2194, ax-12 2215, and ax-13 2406 meeting Conditions (1) and (2). (The "w" suffix stands for "weak version".) Each hypothesis of ax10w 2166, ax11w 2167, and ax12w 2170 is of the form (𝑥 = 𝑦 → (𝜑 ↔ 𝜓)) where 𝜓 is an auxiliary or "dummy" wff metavariable in which 𝑥 doesn't occur. We can show by induction on formula length that the hypotheses can be eliminated in all cases meeting Conditions (1) and (2). The example ax12wdemo 2172 illustrates the techniques (equality theorems and bound variable renaming) used to achieve this. We also show the degenerate instances for axioms with bundled variables in ax11dgen 2168, ax12dgen 2171, ax13dgen1 2174, ax13dgen2 2175, ax13dgen3 2176, and ax13dgen4 2177. (Their proofs are trivial, but we include them to be thorough.) Combining the principal and degenerate cases outside of Metamath, we show that the bundled schemes ax-10 2178, ax-11 2194, ax-12 2215, and ax-13 2406 are schemes of Tarski's system, meaning that all object-language instances they generate are theorems of Tarski's system. It is interesting that Tarski used the bundled scheme ax-6 1990 in an older system, so it seems the main purpose of his later ax6v 1991 was just to show that the weaker unbundled form is sufficient rather than an aesthetic objection to bundled free and bound variables. Since we adopt the bundled ax-6 1990 as our official axiom, we show that the degenerate instance holds in ax6dgen 2165. (Recall that in set.mm, the only statement referencing ax-6 1990 is ax6v 1991.) The case of sp 2221 is curious: originally an axiom scheme of Tarski's system, it was proved logically redundant by Lemma 9 of [KalishMontague] p. 86. However, the proof is by induction on formula length, and the scheme form ∀𝑥𝜑 → 𝜑 apparently cannot be proved directly from Tarski's other axiom schemes. The best we can do seems to be spw 2057, again requiring substitution instances of 𝜑 that meet Conditions (1) and (2) above. Note that our direct proof sp 2221 requires ax-12 2215, which is not part of Tarski's system.
Theorem	ax6dgen 2165	Tarski's system uses the weaker ax6v 1991 instead of the bundled ax-6 1990, so here we show that the degenerate case of ax-6 1990 can be derived. Even though ax-6 1990 is in the list of axioms used, recall that in set.mm, the only statement referencing ax-6 1990 is ax6v 1991. We later rederive from ax6v 1991 the bundled form as ax6 2418 with the help of the auxiliary axiom schemes. (Contributed by NM, 23-Apr-2017.)
⊢ ¬ ∀𝑥 ¬ 𝑥 = 𝑥
Theorem	ax10w 2166*	Weak version of ax-10 2178 from which we can prove any ax-10 2178 instance not involving wff variables or bundling. Uses only Tarski's FOL axiom schemes. It is an alias of hbn1w 2071 introduced for labeling consistency. (Contributed by NM, 9-Apr-2017.) Use hbn1w 2071 instead. (New usage is discouraged.)
⊢ (𝑥 = 𝑦 → (𝜑 ↔ 𝜓))    ⇒   ⊢ (¬ ∀𝑥𝜑 → ∀𝑥 ¬ ∀𝑥𝜑)
Theorem	ax11w 2167*	Weak version of ax-11 2194 from which we can prove any ax-11 2194 instance not involving wff variables or bundling. Uses only Tarski's FOL axiom schemes. Unlike ax-11 2194, this theorem requires that 𝑥 and 𝑦 be distinct i.e. are not bundled. It is an alias of alcomimw 2066 introduced for labeling consistency. (Contributed by NM, 10-Apr-2017.) Use alcomimw 2066 instead. (New usage is discouraged.)
⊢ (𝑦 = 𝑧 → (𝜑 ↔ 𝜓))    ⇒   ⊢ (∀𝑥∀𝑦𝜑 → ∀𝑦∀𝑥𝜑)
Theorem	ax11dgen 2168	Degenerate instance of ax-11 2194 where bundled variables 𝑥 and 𝑦 have a common substitution. Uses only Tarski's FOL axiom schemes. (Contributed by NM, 13-Apr-2017.)
⊢ (∀𝑥∀𝑥𝜑 → ∀𝑥∀𝑥𝜑)
Theorem	ax12wlem 2169*	Lemma for weak version of ax-12 2215. Uses only Tarski's FOL axiom schemes. In some cases, this lemma may lead to shorter proofs than ax12w 2170. (Contributed by NM, 10-Apr-2017.)
⊢ (𝑥 = 𝑦 → (𝜑 ↔ 𝜓))    ⇒   ⊢ (𝑥 = 𝑦 → (𝜑 → ∀𝑥(𝑥 = 𝑦 → 𝜑)))
Theorem	ax12w 2170*	Weak version of ax-12 2215 from which we can prove any ax-12 2215 instance not involving wff variables or bundling. Uses only Tarski's FOL axiom schemes. An instance of the first hypothesis will normally require that 𝑥 and 𝑦 be distinct (unless 𝑥 does not occur in 𝜑). For an example of how the hypotheses can be eliminated when we substitute an expression without wff variables for 𝜑, see ax12wdemo 2172. (Contributed by NM, 10-Apr-2017.)
⊢ (𝑥 = 𝑦 → (𝜑 ↔ 𝜓))    &   ⊢ (𝑦 = 𝑧 → (𝜑 ↔ 𝜒))    ⇒   ⊢ (𝑥 = 𝑦 → (∀𝑦𝜑 → ∀𝑥(𝑥 = 𝑦 → 𝜑)))
Theorem	ax12dgen 2171	Degenerate instance of ax-12 2215 where bundled variables 𝑥 and 𝑦 have a common substitution. Uses only Tarski's FOL axiom schemes. (Contributed by NM, 13-Apr-2017.)
⊢ (𝑥 = 𝑥 → (∀𝑥𝜑 → ∀𝑥(𝑥 = 𝑥 → 𝜑)))
Theorem	ax12wdemo 2172*	Example of an application of ax12w 2170 that results in an instance of ax-12 2215 for a contrived formula with mixed free and bound variables, (𝑥 ∈ 𝑦 ∧ ∀𝑥𝑧 ∈ 𝑥 ∧ ∀𝑦∀𝑧𝑦 ∈ 𝑥), in place of 𝜑. The proof illustrates bound variable renaming with cbvalvw 2059 to obtain fresh variables to avoid distinct variable clashes. Uses only Tarski's FOL axiom schemes. (Contributed by NM, 14-Apr-2017.)
⊢ (𝑥 = 𝑦 → (∀𝑦(𝑥 ∈ 𝑦 ∧ ∀𝑥 𝑧 ∈ 𝑥 ∧ ∀𝑦∀𝑧 𝑦 ∈ 𝑥) → ∀𝑥(𝑥 = 𝑦 → (𝑥 ∈ 𝑦 ∧ ∀𝑥 𝑧 ∈ 𝑥 ∧ ∀𝑦∀𝑧 𝑦 ∈ 𝑥))))
Theorem	ax13w 2173*	Weak version (principal instance) of ax-13 2406. (Because 𝑦 and 𝑧 don't need to be distinct, this actually bundles the principal instance and the degenerate instance (¬ 𝑥 = 𝑦 → (𝑦 = 𝑦 → ∀𝑥𝑦 = 𝑦)).) Uses only Tarski's FOL axiom schemes. The proof is trivial but is included to complete the set ax10w 2166, ax11w 2167, and ax12w 2170. (Contributed by NM, 10-Apr-2017.)
⊢ (¬ 𝑥 = 𝑦 → (𝑦 = 𝑧 → ∀𝑥 𝑦 = 𝑧))
Theorem	ax13dgen1 2174	Degenerate instance of ax-13 2406 where bundled variables 𝑥 and 𝑦 have a common substitution. Uses only Tarski's FOL axiom schemes. (Contributed by NM, 13-Apr-2017.)
⊢ (¬ 𝑥 = 𝑥 → (𝑥 = 𝑧 → ∀𝑥 𝑥 = 𝑧))
Theorem	ax13dgen2 2175	Degenerate instance of ax-13 2406 where bundled variables 𝑥 and 𝑧 have a common substitution. Uses only Tarski's FOL axiom schemes. (Contributed by NM, 13-Apr-2017.)
⊢ (¬ 𝑥 = 𝑦 → (𝑦 = 𝑥 → ∀𝑥 𝑦 = 𝑥))
Theorem	ax13dgen3 2176	Degenerate instance of ax-13 2406 where bundled variables 𝑦 and 𝑧 have a common substitution. Uses only Tarski's FOL axiom schemes. (Contributed by NM, 13-Apr-2017.)
⊢ (¬ 𝑥 = 𝑦 → (𝑦 = 𝑦 → ∀𝑥 𝑦 = 𝑦))
Theorem	ax13dgen4 2177	Degenerate instance of ax-13 2406 where bundled variables 𝑥, 𝑦, and 𝑧 have a common substitution. Therefore, also a degenerate instance of ax13dgen1 2174, ax13dgen2 2175, and ax13dgen3 2176. Also an instance of the intuitionistic tautology pm2.21 124. Uses only Tarski's FOL axiom schemes. (Contributed by NM, 13-Apr-2017.) Reduce axiom usage. (Revised by Wolf Lammen, 10-Oct-2021.)
⊢ (¬ 𝑥 = 𝑥 → (𝑥 = 𝑥 → ∀𝑥 𝑥 = 𝑥))
1.5  Predicate calculus with equality: Auxiliary axiom schemes (4 schemes) In this section we introduce four additional schemes ax-10 2178, ax-11 2194, ax-12 2215, and ax-13 2406 that are not part of Tarski's system but can be proved (outside of Metamath) as theorem schemes of Tarski's system. These are needed to give our system the property of "scheme completeness", which means that we can prove (with Metamath) all possible theorem schemes expressible in our language of wff metavariables ranging over object-language wffs, and setvar variables ranging over object-language individual variables. To show that these schemes are valid metatheorems of Tarski's system S2, above we proved from Tarski's system theorems ax10w 2166, ax11w 2167, ax12w 2170, and ax13w 2173, which show that any object-language instance of these schemes (emulated by having no wff metavariables and requiring all setvar variables to be mutually distinct) can be proved using only the schemes in Tarski's system S2. An open problem is to show that these four additional schemes are mutually metalogically independent and metalogically independent from Tarski's. So far, independence of ax-12 2215 from all others has been shown, and independence of Tarski's ax-6 1990 from all others has been shown; see items 9a and 11 on https://us.metamath.org/award2003.html 1990.
1.5.1  Axiom scheme ax-10 (Quantified Negation)
Axiom	ax-10 2178	Axiom of Quantified Negation. Axiom C5-2 of [Monk2] p. 113. This axiom scheme is logically redundant (see ax10w 2166) but is used as an auxiliary axiom scheme to achieve scheme completeness. It means that 𝑥 is not free in ¬ ∀𝑥𝜑. (Contributed by NM, 21-May-2008.) Use its alias hbn1 2179 instead if you must use it. Any theorem in first-order logic (FOL) that contains only set variables that are all mutually distinct, and has no wff variables, can be proved *without* using ax-10 2178 through ax-13 2406, by invoking ax10w 2166 through ax13w 2173. We encourage proving theorems *without* ax-10 2178 through ax-13 2406 and moving them up to the ax-4 1832 through ax-9 2155 section. (New usage is discouraged.)
⊢ (¬ ∀𝑥𝜑 → ∀𝑥 ¬ ∀𝑥𝜑)
Theorem	hbn1 2179	Alias for ax-10 2178 to be used instead of it. (Contributed by NM, 24-Jan-1993.) (Proof shortened by Wolf Lammen, 18-Aug-2014.)
⊢ (¬ ∀𝑥𝜑 → ∀𝑥 ¬ ∀𝑥𝜑)
Theorem	hbe1 2180	The setvar 𝑥 is not free in ∃𝑥𝜑. Corresponds to the axiom (5) of modal logic (see also modal5 2192). (Contributed by NM, 24-Jan-1993.)
⊢ (∃𝑥𝜑 → ∀𝑥∃𝑥𝜑)
Theorem	hbe1a 2181	Dual statement of hbe1 2180. Modified version of axc7e 2353 with a universally quantified consequent. (Contributed by Wolf Lammen, 15-Sep-2021.)
⊢ (∃𝑥∀𝑥𝜑 → ∀𝑥𝜑)
Theorem	nf5-1 2182	One direction of nf5 2319 can be proved with a smaller footprint on axiom usage. (Contributed by Wolf Lammen, 16-Sep-2021.)
⊢ (∀𝑥(𝜑 → ∀𝑥𝜑) → Ⅎ𝑥𝜑)
Theorem	nf5i 2183	Deduce that 𝑥 is not free in 𝜑 from the definition. (Contributed by Mario Carneiro, 11-Aug-2016.)
⊢ (𝜑 → ∀𝑥𝜑)    ⇒   ⊢ Ⅎ𝑥𝜑
Theorem	nf5dh 2184	Deduce that 𝑥 is not free in 𝜓 in a context. (Contributed by Mario Carneiro, 24-Sep-2016.) df-nf 1807 changed. (Revised by Wolf Lammen, 11-Oct-2021.)
⊢ (𝜑 → ∀𝑥𝜑)    &   ⊢ (𝜑 → (𝜓 → ∀𝑥𝜓))    ⇒   ⊢ (𝜑 → Ⅎ𝑥𝜓)
Theorem	nf5dv 2185*	Apply the definition of not-free in a context. (Contributed by Mario Carneiro, 11-Aug-2016.) df-nf 1807 changed. (Revised by Wolf Lammen, 18-Sep-2021.) (Proof shortened by Wolf Lammen, 13-Jul-2022.)
⊢ (𝜑 → (𝜓 → ∀𝑥𝜓))    ⇒   ⊢ (𝜑 → Ⅎ𝑥𝜓)
Theorem	nfnaew 2186*	All variables are effectively bound in a distinct variable specifier. Version of nfnae 2468 with a disjoint variable condition, which does not require ax-13 2406. (Contributed by Mario Carneiro, 11-Aug-2016.) Avoid ax-13 2406. (Revised by GG, 10-Jan-2024.) (Proof shortened by Wolf Lammen, 25-Sep-2024.)
⊢ Ⅎ𝑧 ¬ ∀𝑥 𝑥 = 𝑦
Theorem	nfe1 2187	The setvar 𝑥 is not free in ∃𝑥𝜑. (Contributed by Mario Carneiro, 11-Aug-2016.)
⊢ Ⅎ𝑥∃𝑥𝜑
Theorem	nfa1 2188	The setvar 𝑥 is not free in ∀𝑥𝜑. (Contributed by Mario Carneiro, 11-Aug-2016.) df-nf 1807 changed. (Revised by Wolf Lammen, 11-Sep-2021.) Remove dependency on ax-12 2215. (Revised by Wolf Lammen, 12-Oct-2021.)
⊢ Ⅎ𝑥∀𝑥𝜑
Theorem	nfna1 2189	A convenience theorem particularly designed to remove dependencies on ax-11 2194 in conjunction with distinctors. (Contributed by Wolf Lammen, 2-Sep-2018.)
⊢ Ⅎ𝑥 ¬ ∀𝑥𝜑
Theorem	nfia1 2190	Lemma 23 of [Monk2] p. 114. (Contributed by Mario Carneiro, 24-Sep-2016.)
⊢ Ⅎ𝑥(∀𝑥𝜑 → ∀𝑥𝜓)
Theorem	nfnf1 2191	The setvar 𝑥 is not free in Ⅎ𝑥𝜑. (Contributed by Mario Carneiro, 11-Aug-2016.) Remove dependency on ax-12 2215. (Revised by Wolf Lammen, 12-Oct-2021.)
⊢ Ⅎ𝑥Ⅎ𝑥𝜑
Theorem	modal5 2192	The analogue in our predicate calculus of axiom (5) of modal logic S5. See also hbe1 2180. (Contributed by NM, 5-Oct-2005.)
⊢ (¬ ∀𝑥 ¬ 𝜑 → ∀𝑥 ¬ ∀𝑥 ¬ 𝜑)
Theorem	nfs1v 2193*	The setvar 𝑥 is not free in [𝑦 / 𝑥]𝜑 when 𝑥 and 𝑦 are distinct. (Contributed by Mario Carneiro, 11-Aug-2016.) Shorten nfs1v 2193 and hbs1 2311 combined. (Revised by Wolf Lammen, 28-Jul-2022.)
⊢ Ⅎ𝑥[𝑦 / 𝑥]𝜑
1.5.2  Axiom scheme ax-11 (Quantifier Commutation)
Axiom	ax-11 2194	Axiom of Quantifier Commutation. This axiom says universal quantifiers can be swapped. Axiom scheme C6' in [Megill] p. 448 (p. 16 of the preprint). Also appears as Lemma 12 of [Monk2] p. 109 and Axiom C5-3 of [Monk2] p. 113. This axiom scheme is logically redundant (see ax11w 2167) but is used as an auxiliary axiom scheme to achieve metalogical completeness. Use its weak version alcomimw 2066 when it allows to avoid dependence on ax-11 2194. (Contributed by NM, 12-Mar-1993.)
⊢ (∀𝑥∀𝑦𝜑 → ∀𝑦∀𝑥𝜑)
Theorem	alcoms 2195	Swap quantifiers in an antecedent. (Contributed by NM, 11-May-1993.)
⊢ (∀𝑥∀𝑦𝜑 → 𝜓)    ⇒   ⊢ (∀𝑦∀𝑥𝜑 → 𝜓)
Theorem	alcom 2196	Theorem 19.5 of [Margaris] p. 89. Use its weak version alcomw 2068 when it allows to avoid dependence on ax-11 2194. (Contributed by NM, 30-Jun-1993.)
⊢ (∀𝑥∀𝑦𝜑 ↔ ∀𝑦∀𝑥𝜑)
Theorem	alrot3 2197	Theorem *11.21 in [WhiteheadRussell] p. 160. (Contributed by Andrew Salmon, 24-May-2011.)
⊢ (∀𝑥∀𝑦∀𝑧𝜑 ↔ ∀𝑦∀𝑧∀𝑥𝜑)
Theorem	alrot4 2198	Rotate four universal quantifiers twice. (Contributed by NM, 2-Feb-2005.) (Proof shortened by Fan Zheng, 6-Jun-2016.)
⊢ (∀𝑥∀𝑦∀𝑧∀𝑤𝜑 ↔ ∀𝑧∀𝑤∀𝑥∀𝑦𝜑)
Theorem	excom 2199	Theorem 19.11 of [Margaris] p. 89. (Contributed by NM, 5-Aug-1993.) Remove dependencies on ax-5 1933, ax-6 1990, ax-7 2031, ax-10 2178, ax-12 2215. (Revised by Wolf Lammen, 8-Jan-2018.) (Proof shortened by Wolf Lammen, 22-Aug-2020.)
⊢ (∃𝑥∃𝑦𝜑 ↔ ∃𝑦∃𝑥𝜑)
Theorem	excomim 2200	One direction of Theorem 19.11 of [Margaris] p. 89. (Contributed by NM, 5-Aug-1993.) (Revised by Mario Carneiro, 24-Sep-2016.) Remove dependencies on ax-5 1933, ax-6 1990, ax-7 2031, ax-10 2178, ax-12 2215. (Revised by Wolf Lammen, 8-Jan-2018.)
⊢ (∃𝑥∃𝑦𝜑 → ∃𝑦∃𝑥𝜑)