P. 15 of Theorem List - Intuitionistic Logic Explorer

Theorem List for Intuitionistic Logic Explorer - 1401-1500   *Has distinct variable group(s)

Type	Label	Description
Statement
Theorem	alrimih 1401	Inference from Theorem 19.21 of [Margaris] p. 90. (Contributed by NM, 5-Aug-1993.) (New usage is discouraged.)
⊢ (𝜑 → ∀𝑥𝜑)    &   ⊢ (𝜑 → 𝜓)    ⇒   ⊢ (𝜑 → ∀𝑥𝜓)
Theorem	albii 1402	Inference adding universal quantifier to both sides of an equivalence. (Contributed by NM, 7-Aug-1994.)
⊢ (𝜑 ↔ 𝜓)    ⇒   ⊢ (∀𝑥𝜑 ↔ ∀𝑥𝜓)
Theorem	2albii 1403	Inference adding 2 universal quantifiers to both sides of an equivalence. (Contributed by NM, 9-Mar-1997.)
⊢ (𝜑 ↔ 𝜓)    ⇒   ⊢ (∀𝑥∀𝑦𝜑 ↔ ∀𝑥∀𝑦𝜓)
Theorem	hbxfrbi 1404	A utility lemma to transfer a bound-variable hypothesis builder into a definition. (Contributed by Jonathan Ben-Naim, 3-Jun-2011.)
⊢ (𝜑 ↔ 𝜓)    &   ⊢ (𝜓 → ∀𝑥𝜓)    ⇒   ⊢ (𝜑 → ∀𝑥𝜑)
Theorem	nfbii 1405	Equality theorem for not-free. (Contributed by Mario Carneiro, 11-Aug-2016.)
⊢ (𝜑 ↔ 𝜓)    ⇒   ⊢ (Ⅎ𝑥𝜑 ↔ Ⅎ𝑥𝜓)
Theorem	nfxfr 1406	A utility lemma to transfer a bound-variable hypothesis builder into a definition. (Contributed by Mario Carneiro, 11-Aug-2016.)
⊢ (𝜑 ↔ 𝜓)    &   ⊢ Ⅎ𝑥𝜓    ⇒   ⊢ Ⅎ𝑥𝜑
Theorem	nfxfrd 1407	A utility lemma to transfer a bound-variable hypothesis builder into a definition. (Contributed by Mario Carneiro, 24-Sep-2016.)
⊢ (𝜑 ↔ 𝜓)    &   ⊢ (𝜒 → Ⅎ𝑥𝜓)    ⇒   ⊢ (𝜒 → Ⅎ𝑥𝜑)
Theorem	alcoms 1408	Swap quantifiers in an antecedent. (Contributed by NM, 11-May-1993.)
⊢ (∀𝑥∀𝑦𝜑 → 𝜓)    ⇒   ⊢ (∀𝑦∀𝑥𝜑 → 𝜓)
Theorem	hbal 1409	If 𝑥 is not free in 𝜑, it is not free in ∀𝑦𝜑. (Contributed by NM, 5-Aug-1993.)
⊢ (𝜑 → ∀𝑥𝜑)    ⇒   ⊢ (∀𝑦𝜑 → ∀𝑥∀𝑦𝜑)
Theorem	alcom 1410	Theorem 19.5 of [Margaris] p. 89. (Contributed by NM, 5-Aug-1993.)
⊢ (∀𝑥∀𝑦𝜑 ↔ ∀𝑦∀𝑥𝜑)
Theorem	alrimdh 1411	Deduction from Theorem 19.21 of [Margaris] p. 90. (Contributed by NM, 10-Feb-1997.) (Proof shortened by Andrew Salmon, 13-May-2011.)
⊢ (𝜑 → ∀𝑥𝜑)    &   ⊢ (𝜓 → ∀𝑥𝜓)    &   ⊢ (𝜑 → (𝜓 → 𝜒))    ⇒   ⊢ (𝜑 → (𝜓 → ∀𝑥𝜒))
Theorem	albidh 1412	Formula-building rule for universal quantifier (deduction rule). (Contributed by NM, 5-Aug-1993.)
⊢ (𝜑 → ∀𝑥𝜑)    &   ⊢ (𝜑 → (𝜓 ↔ 𝜒))    ⇒   ⊢ (𝜑 → (∀𝑥𝜓 ↔ ∀𝑥𝜒))
Theorem	19.26 1413	Theorem 19.26 of [Margaris] p. 90. Also Theorem *10.22 of [WhiteheadRussell] p. 119. (Contributed by NM, 5-Aug-1993.) (Proof shortened by Wolf Lammen, 4-Jul-2014.)
⊢ (∀𝑥(𝜑 ∧ 𝜓) ↔ (∀𝑥𝜑 ∧ ∀𝑥𝜓))
Theorem	19.26-2 1414	Theorem 19.26 of [Margaris] p. 90 with two quantifiers. (Contributed by NM, 3-Feb-2005.)
⊢ (∀𝑥∀𝑦(𝜑 ∧ 𝜓) ↔ (∀𝑥∀𝑦𝜑 ∧ ∀𝑥∀𝑦𝜓))
Theorem	19.26-3an 1415	Theorem 19.26 of [Margaris] p. 90 with triple conjunction. (Contributed by NM, 13-Sep-2011.)
⊢ (∀𝑥(𝜑 ∧ 𝜓 ∧ 𝜒) ↔ (∀𝑥𝜑 ∧ ∀𝑥𝜓 ∧ ∀𝑥𝜒))
Theorem	19.33 1416	Theorem 19.33 of [Margaris] p. 90. (Contributed by NM, 5-Aug-1993.)
⊢ ((∀𝑥𝜑 ∨ ∀𝑥𝜓) → ∀𝑥(𝜑 ∨ 𝜓))
Theorem	alrot3 1417	Theorem *11.21 in [WhiteheadRussell] p. 160. (Contributed by Andrew Salmon, 24-May-2011.)
⊢ (∀𝑥∀𝑦∀𝑧𝜑 ↔ ∀𝑦∀𝑧∀𝑥𝜑)
Theorem	alrot4 1418	Rotate 4 universal quantifiers twice. (Contributed by NM, 2-Feb-2005.) (Proof shortened by Wolf Lammen, 28-Jun-2014.)
⊢ (∀𝑥∀𝑦∀𝑧∀𝑤𝜑 ↔ ∀𝑧∀𝑤∀𝑥∀𝑦𝜑)
Theorem	albiim 1419	Split a biconditional and distribute quantifier. (Contributed by NM, 18-Aug-1993.)
⊢ (∀𝑥(𝜑 ↔ 𝜓) ↔ (∀𝑥(𝜑 → 𝜓) ∧ ∀𝑥(𝜓 → 𝜑)))
Theorem	2albiim 1420	Split a biconditional and distribute 2 quantifiers. (Contributed by NM, 3-Feb-2005.)
⊢ (∀𝑥∀𝑦(𝜑 ↔ 𝜓) ↔ (∀𝑥∀𝑦(𝜑 → 𝜓) ∧ ∀𝑥∀𝑦(𝜓 → 𝜑)))
Theorem	hband 1421	Deduction form of bound-variable hypothesis builder hban 1482. (Contributed by NM, 2-Jan-2002.)
⊢ (𝜑 → (𝜓 → ∀𝑥𝜓))    &   ⊢ (𝜑 → (𝜒 → ∀𝑥𝜒))    ⇒   ⊢ (𝜑 → ((𝜓 ∧ 𝜒) → ∀𝑥(𝜓 ∧ 𝜒)))
Theorem	hb3and 1422	Deduction form of bound-variable hypothesis builder hb3an 1485. (Contributed by NM, 17-Feb-2013.)
⊢ (𝜑 → (𝜓 → ∀𝑥𝜓))    &   ⊢ (𝜑 → (𝜒 → ∀𝑥𝜒))    &   ⊢ (𝜑 → (𝜃 → ∀𝑥𝜃))    ⇒   ⊢ (𝜑 → ((𝜓 ∧ 𝜒 ∧ 𝜃) → ∀𝑥(𝜓 ∧ 𝜒 ∧ 𝜃)))
Theorem	hbald 1423	Deduction form of bound-variable hypothesis builder hbal 1409. (Contributed by NM, 2-Jan-2002.)
⊢ (𝜑 → ∀𝑦𝜑)    &   ⊢ (𝜑 → (𝜓 → ∀𝑥𝜓))    ⇒   ⊢ (𝜑 → (∀𝑦𝜓 → ∀𝑥∀𝑦𝜓))
Syntax	wex 1424	Extend wff definition to include the existential quantifier ("there exists").
wff ∃𝑥𝜑
Axiom	ax-ie1 1425	𝑥 is bound in ∃𝑥𝜑. One of the axioms of predicate logic. (Contributed by Mario Carneiro, 31-Jan-2015.)
⊢ (∃𝑥𝜑 → ∀𝑥∃𝑥𝜑)
Axiom	ax-ie2 1426	Define existential quantification. ∃𝑥𝜑 means "there exists at least one set 𝑥 such that 𝜑 is true." One of the axioms of predicate logic. (Contributed by Mario Carneiro, 31-Jan-2015.)
⊢ (∀𝑥(𝜓 → ∀𝑥𝜓) → (∀𝑥(𝜑 → 𝜓) ↔ (∃𝑥𝜑 → 𝜓)))
Theorem	hbe1 1427	𝑥 is not free in ∃𝑥𝜑. (Contributed by NM, 5-Aug-1993.)
⊢ (∃𝑥𝜑 → ∀𝑥∃𝑥𝜑)
Theorem	nfe1 1428	𝑥 is not free in ∃𝑥𝜑. (Contributed by Mario Carneiro, 11-Aug-2016.)
⊢ Ⅎ𝑥∃𝑥𝜑
Theorem	19.23ht 1429	Closed form of Theorem 19.23 of [Margaris] p. 90. (Contributed by NM, 7-Nov-2005.) (Revised by Mario Carneiro, 1-Feb-2015.)
⊢ (∀𝑥(𝜓 → ∀𝑥𝜓) → (∀𝑥(𝜑 → 𝜓) ↔ (∃𝑥𝜑 → 𝜓)))
Theorem	19.23h 1430	Theorem 19.23 of [Margaris] p. 90. (Contributed by NM, 5-Aug-1993.) (Revised by Mario Carneiro, 1-Feb-2015.)
⊢ (𝜓 → ∀𝑥𝜓)    ⇒   ⊢ (∀𝑥(𝜑 → 𝜓) ↔ (∃𝑥𝜑 → 𝜓))
Theorem	alnex 1431	Theorem 19.7 of [Margaris] p. 89. To read this intuitionistically, think of it as "if 𝜑 can be refuted for all 𝑥, then it is not possible to find an 𝑥 for which 𝜑 holds" (and likewise for the converse). Comparing this with dfexdc 1433 illustrates that statements which look similar (to someone used to classical logic) can be different intuitionistically due to different placement of negations. (Contributed by NM, 5-Aug-1993.) (Revised by NM, 1-Feb-2015.) (Revised by Mario Carneiro, 12-May-2015.)
⊢ (∀𝑥 ¬ 𝜑 ↔ ¬ ∃𝑥𝜑)
Theorem	nex 1432	Generalization rule for negated wff. (Contributed by NM, 18-May-1994.)
⊢ ¬ 𝜑    ⇒   ⊢ ¬ ∃𝑥𝜑
Theorem	dfexdc 1433	Defining ∃𝑥𝜑 given decidability. It is common in classical logic to define ∃𝑥𝜑 as ¬ ∀𝑥¬ 𝜑 but in intuitionistic logic without a decidability condition, that is only an implication not an equivalence, as seen at exalim 1434. (Contributed by Jim Kingdon, 15-Mar-2018.)
⊢ (DECID ∃𝑥𝜑 → (∃𝑥𝜑 ↔ ¬ ∀𝑥 ¬ 𝜑))
Theorem	exalim 1434	One direction of a classical definition of existential quantification. One direction of Definition of [Margaris] p. 49. For a decidable proposition, this is an equivalence, as seen as dfexdc 1433. (Contributed by Jim Kingdon, 29-Jul-2018.)
⊢ (∃𝑥𝜑 → ¬ ∀𝑥 ¬ 𝜑)
1.3.2  Equality predicate (continued) The equality predicate was introduced above in wceq 1287 for use by df-tru 1290. See the comments in that section. In this section, we continue with the first "real" use of it.
Theorem	weq 1435	Extend wff definition to include atomic formulas using the equality predicate. (Instead of introducing weq 1435 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 wceq 1287. This lets us avoid overloading the = connective, thus preventing ambiguity that would complicate certain Metamath parsers. However, logically weq 1435 is considered to be a primitive syntax, even though here it is artificially "derived" from wceq 1287. Note: To see the proof steps of this syntax proof, type "show proof weq /all" in the Metamath program.) (Contributed by NM, 24-Jan-2006.)
wff 𝑥 = 𝑦
Syntax	wcel 1436	Extend wff definition to include the membership connective between classes. (The purpose of introducing wff 𝐴 ∈ 𝐵 here is to allow us to express i.e. "prove" the wel 1437 of predicate calculus in terms of the wceq 1287 of set theory, so that we don't "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.)
wff 𝐴 ∈ 𝐵
Theorem	wel 1437	Extend wff definition to include atomic formulas with the epsilon (membership) predicate. This is read "𝑥 is an element of 𝑦," "𝑥 is a member of 𝑦," "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 syntactical construction introduces a binary non-logical predicate symbol ∈ (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 1437 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 1436. This lets us avoid overloading the ∈ connective, thus preventing ambiguity that would complicate certain Metamath parsers. However, logically wel 1437 is considered to be a primitive syntax, even though here it is artificially "derived" from wcel 1436. Note: To see the proof steps of this syntax proof, type "show proof wel /all" in the Metamath program.) (Contributed by NM, 24-Jan-2006.)
wff 𝑥 ∈ 𝑦
Axiom	ax-8 1438	Axiom of Equality. One of the equality and substitution axioms of predicate calculus with equality. This is similar to, but not quite, a transitive law for equality (proved later as equtr 1639). Axiom scheme C8' in [Megill] p. 448 (p. 16 of the preprint). Also appears as Axiom C7 of [Monk2] p. 105. Axioms ax-8 1438 through ax-16 1739 are the axioms having to do with equality, substitution, and logical properties of our binary predicate ∈ (which later in set theory will mean "is a member of"). Note that all axioms except ax-16 1739 and ax-17 1462 are still valid even when 𝑥, 𝑦, and 𝑧 are replaced with the same variable because they do not have any distinct variable (Metamath's $d) restrictions. Distinct variable restrictions are required for ax-16 1739 and ax-17 1462 only. (Contributed by NM, 5-Aug-1993.)
⊢ (𝑥 = 𝑦 → (𝑥 = 𝑧 → 𝑦 = 𝑧))
Axiom	ax-10 1439	Axiom of Quantifier Substitution. One of the equality and substitution axioms of predicate calculus with equality. Appears as Lemma L12 in [Megill] p. 445 (p. 12 of the preprint). The original version of this axiom was ax-10o 1648 ("o" for "old") and was replaced with this shorter ax-10 1439 in May 2008. The old axiom is proved from this one as theorem ax10o 1647. Conversely, this axiom is proved from ax-10o 1648 as theorem ax10 1649. (Contributed by NM, 5-Aug-1993.)
⊢ (∀𝑥 𝑥 = 𝑦 → ∀𝑦 𝑦 = 𝑥)
Axiom	ax-11 1440	Axiom of Variable 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 1811). 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. Variants of this axiom which are equivalent in classical logic but which have not been shown to be equivalent for intuitionistic logic are ax11v 1752, ax11v2 1745 and ax-11o 1748. (Contributed by NM, 5-Aug-1993.)
⊢ (𝑥 = 𝑦 → (∀𝑦𝜑 → ∀𝑥(𝑥 = 𝑦 → 𝜑)))
Axiom	ax-i12 1441	Axiom of Quantifier Introduction. One of the equality and substitution axioms of predicate calculus with equality. Informally, it says that whenever 𝑧 is distinct from 𝑥 and 𝑦, and 𝑥 = 𝑦 is true, then 𝑥 = 𝑦 quantified with 𝑧 is also true. In other words, 𝑧 is irrelevant to the truth of 𝑥 = 𝑦. Axiom scheme C9' in [Megill] p. 448 (p. 16 of the preprint). It apparently does not otherwise appear in the literature but is easily proved from textbook predicate calculus by cases. This axiom has been modified from the original ax-12 1445 for compatibility with intuitionistic logic. (Contributed by Mario Carneiro, 31-Jan-2015.)
⊢ (∀𝑧 𝑧 = 𝑥 ∨ (∀𝑧 𝑧 = 𝑦 ∨ ∀𝑧(𝑥 = 𝑦 → ∀𝑧 𝑥 = 𝑦)))
Axiom	ax-bndl 1442	Axiom of bundling. The general idea of this axiom is that two variables are either distinct or non-distinct. That idea could be expressed as ∀𝑧𝑧 = 𝑥 ∨ ¬ ∀𝑧𝑧 = 𝑥. However, we instead choose an axiom which has many of the same consequences, but which is different with respect to a universe which contains only one object. ∀𝑧𝑧 = 𝑥 holds if 𝑧 and 𝑥 are the same variable, likewise for 𝑧 and 𝑦, and ∀𝑥∀𝑧(𝑥 = 𝑦 → ∀𝑧𝑥 = 𝑦) holds if 𝑧 is distinct from the others (and the universe has at least two objects). As with other statements of the form "x is decidable (either true or false)", this does not entail the full Law of the Excluded Middle (which is the proposition that all statements are decidable), but instead merely the assertion that particular kinds of statements are decidable (or in this case, an assertion similar to decidability). This axiom implies ax-i12 1441 as can be seen at axi12 1450. Whether ax-bndl 1442 can be proved from the remaining axioms including ax-i12 1441 is not known. The reason we call this "bundling" is that a statement without a distinct variable constraint "bundles" together two statements, one in which the two variables are the same and one in which they are different. (Contributed by Mario Carneiro and Jim Kingdon, 14-Mar-2018.)
⊢ (∀𝑧 𝑧 = 𝑥 ∨ (∀𝑧 𝑧 = 𝑦 ∨ ∀𝑥∀𝑧(𝑥 = 𝑦 → ∀𝑧 𝑥 = 𝑦)))
Axiom	ax-4 1443	Axiom of Specialization. A quantified wff implies the wff without a quantifier (i.e. an instance, or special case, of the generalized wff). In other words if something is true for all 𝑥, it is true for any specific 𝑥 (that would typically occur as a free variable in the wff substituted for 𝜑). (A free variable is one that does not occur in the scope of a quantifier: 𝑥 and 𝑦 are both free in 𝑥 = 𝑦, but only 𝑥 is free in ∀𝑦𝑥 = 𝑦.) Axiom scheme C5' in [Megill] p. 448 (p. 16 of the preprint). Also appears as Axiom B5 of [Tarski] p. 67 (under his system S2, defined in the last paragraph on p. 77). Note that the converse of this axiom does not hold in general, but a weaker inference form of the converse holds and is expressed as rule ax-gen 1381. Conditional forms of the converse are given by ax-12 1445, ax-16 1739, and ax-17 1462. Unlike the more general textbook Axiom of Specialization, we cannot choose a variable different from 𝑥 for the special case. For use, that requires the assistance of equality axioms, and we deal with it later after we introduce the definition of proper substitution - see stdpc4 1702. (Contributed by NM, 5-Aug-1993.)
⊢ (∀𝑥𝜑 → 𝜑)
Theorem	sp 1444	Specialization. Another name for ax-4 1443. (Contributed by NM, 21-May-2008.)
⊢ (∀𝑥𝜑 → 𝜑)
Theorem	ax-12 1445	Rederive the original version of the axiom from ax-i12 1441. (Contributed by Mario Carneiro, 3-Feb-2015.)
⊢ (¬ ∀𝑧 𝑧 = 𝑥 → (¬ ∀𝑧 𝑧 = 𝑦 → (𝑥 = 𝑦 → ∀𝑧 𝑥 = 𝑦)))
Theorem	ax12or 1446	Another name for ax-i12 1441. (Contributed by NM, 3-Feb-2015.)
⊢ (∀𝑧 𝑧 = 𝑥 ∨ (∀𝑧 𝑧 = 𝑦 ∨ ∀𝑧(𝑥 = 𝑦 → ∀𝑧 𝑥 = 𝑦)))
Axiom	ax-13 1447	Axiom of Equality. 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 the ∈ binary predicate. Axiom scheme C12' in [Megill] p. 448 (p. 16 of the preprint). It is a special case of Axiom B8 (p. 75) of system S2 of [Tarski] p. 77. "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. (Contributed by NM, 5-Aug-1993.)
⊢ (𝑥 = 𝑦 → (𝑥 ∈ 𝑧 → 𝑦 ∈ 𝑧))
Axiom	ax-14 1448	Axiom of Equality. 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 the ∈ binary predicate. Axiom scheme C13' in [Megill] p. 448 (p. 16 of the preprint). It is a special case of Axiom B8 (p. 75) of system S2 of [Tarski] p. 77. (Contributed by NM, 5-Aug-1993.)
⊢ (𝑥 = 𝑦 → (𝑧 ∈ 𝑥 → 𝑧 ∈ 𝑦))
Theorem	hbequid 1449	Bound-variable hypothesis builder for 𝑥 = 𝑥. This theorem tells us that any variable, including 𝑥, is effectively not free in 𝑥 = 𝑥, even though 𝑥 is technically free according to the traditional definition of free variable. (The proof uses only ax-5 1379, ax-8 1438, ax-12 1445, and ax-gen 1381. This shows that this can be proved without ax-9 1467, even though the theorem equid 1632 cannot be. A shorter proof using ax-9 1467 is obtainable from equid 1632 and hbth 1395.) (Contributed by NM, 13-Jan-2011.) (Proof shortened by Wolf Lammen, 23-Mar-2014.)
⊢ (𝑥 = 𝑥 → ∀𝑦 𝑥 = 𝑥)
Theorem	axi12 1450	Proof that ax-i12 1441 follows from ax-bndl 1442. So that we can track which theorems rely on ax-bndl 1442, proofs should reference ax-i12 1441 rather than this theorem. (Contributed by Jim Kingdon, 17-Aug-2018.) (New usage is discouraged). (Proof modification is discouraged.)
⊢ (∀𝑧 𝑧 = 𝑥 ∨ (∀𝑧 𝑧 = 𝑦 ∨ ∀𝑧(𝑥 = 𝑦 → ∀𝑧 𝑥 = 𝑦)))
Theorem	alequcom 1451	Commutation law for identical variable specifiers. The antecedent and consequent are true when 𝑥 and 𝑦 are substituted with the same variable. Lemma L12 in [Megill] p. 445 (p. 12 of the preprint). (Contributed by NM, 5-Aug-1993.)
⊢ (∀𝑥 𝑥 = 𝑦 → ∀𝑦 𝑦 = 𝑥)
Theorem	alequcoms 1452	A commutation rule for identical variable specifiers. (Contributed by NM, 5-Aug-1993.)
⊢ (∀𝑥 𝑥 = 𝑦 → 𝜑)    ⇒   ⊢ (∀𝑦 𝑦 = 𝑥 → 𝜑)
Theorem	nalequcoms 1453	A commutation rule for distinct variable specifiers. (Contributed by NM, 2-Jan-2002.) (Revised by Mario Carneiro, 2-Feb-2015.)
⊢ (¬ ∀𝑥 𝑥 = 𝑦 → 𝜑)    ⇒   ⊢ (¬ ∀𝑦 𝑦 = 𝑥 → 𝜑)
Theorem	nfr 1454	Consequence of the definition of not-free. (Contributed by Mario Carneiro, 26-Sep-2016.)
⊢ (Ⅎ𝑥𝜑 → (𝜑 → ∀𝑥𝜑))
Theorem	nfri 1455	Consequence of the definition of not-free. (Contributed by Mario Carneiro, 11-Aug-2016.)
⊢ Ⅎ𝑥𝜑    ⇒   ⊢ (𝜑 → ∀𝑥𝜑)
Theorem	nfrd 1456	Consequence of the definition of not-free in a context. (Contributed by Mario Carneiro, 11-Aug-2016.)
⊢ (𝜑 → Ⅎ𝑥𝜓)    ⇒   ⊢ (𝜑 → (𝜓 → ∀𝑥𝜓))
Theorem	alimd 1457	Deduction from Theorem 19.20 of [Margaris] p. 90. (Contributed by Mario Carneiro, 24-Sep-2016.)
⊢ Ⅎ𝑥𝜑    &   ⊢ (𝜑 → (𝜓 → 𝜒))    ⇒   ⊢ (𝜑 → (∀𝑥𝜓 → ∀𝑥𝜒))
Theorem	alrimi 1458	Inference from Theorem 19.21 of [Margaris] p. 90. (Contributed by Mario Carneiro, 24-Sep-2016.)
⊢ Ⅎ𝑥𝜑    &   ⊢ (𝜑 → 𝜓)    ⇒   ⊢ (𝜑 → ∀𝑥𝜓)
Theorem	nfd 1459	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	nfdh 1460	Deduce that 𝑥 is not free in 𝜓 in a context. (Contributed by Mario Carneiro, 24-Sep-2016.)
⊢ (𝜑 → ∀𝑥𝜑)    &   ⊢ (𝜑 → (𝜓 → ∀𝑥𝜓))    ⇒   ⊢ (𝜑 → Ⅎ𝑥𝜓)
Theorem	nfrimi 1461	Moving an antecedent outside Ⅎ. (Contributed by Jim Kingdon, 23-Mar-2018.)
⊢ Ⅎ𝑥𝜑    &   ⊢ Ⅎ𝑥(𝜑 → 𝜓)    ⇒   ⊢ (𝜑 → Ⅎ𝑥𝜓)
1.3.3  Axiom ax-17 - first use of the $d distinct variable statement
Axiom	ax-17 1462*	Axiom to quantify a variable over a formula in which it does not occur. Axiom C5 in [Megill] p. 444 (p. 11 of the preprint). Also appears as Axiom B6 (p. 75) of system S2 of [Tarski] p. 77 and Axiom C5-1 of [Monk2] p. 113. (Contributed by NM, 5-Aug-1993.)
⊢ (𝜑 → ∀𝑥𝜑)
Theorem	a17d 1463*	ax-17 1462 with antecedent. (Contributed by NM, 1-Mar-2013.)
⊢ (𝜑 → (𝜓 → ∀𝑥𝜓))
Theorem	nfv 1464*	If 𝑥 is not present in 𝜑, then 𝑥 is not free in 𝜑. (Contributed by Mario Carneiro, 11-Aug-2016.)
⊢ Ⅎ𝑥𝜑
Theorem	nfvd 1465*	nfv 1464 with antecedent. Useful in proofs of deduction versions of bound-variable hypothesis builders such as nfimd 1520. (Contributed by Mario Carneiro, 6-Oct-2016.)
⊢ (𝜑 → Ⅎ𝑥𝜓)
1.3.4  Introduce Axiom of Existence
Axiom	ax-i9 1466	Axiom of Existence. One of the equality and substitution axioms of predicate calculus with equality. One thing this axiom tells us is that at least one thing exists (although ax-4 1443 and possibly others also tell us that, i.e. they are not valid in the empty domain of a "free logic"). In this form (not requiring that 𝑥 and 𝑦 be distinct) it was used in an axiom system of Tarski (see Axiom B7' in footnote 1 of [KalishMontague] p. 81.) Another name for this theorem is a9e 1629, which has additional remarks. (Contributed by Mario Carneiro, 31-Jan-2015.)
⊢ ∃𝑥 𝑥 = 𝑦
Theorem	ax-9 1467	Derive ax-9 1467 from ax-i9 1466, the modified version for intuitionistic logic. Although ax-9 1467 does hold intuistionistically, in intuitionistic logic it is weaker than ax-i9 1466. (Contributed by NM, 3-Feb-2015.)
⊢ ¬ ∀𝑥 ¬ 𝑥 = 𝑦
Theorem	equidqe 1468	equid 1632 with some quantification and negation without using ax-4 1443 or ax-17 1462. (Contributed by NM, 13-Jan-2011.) (Proof shortened by Wolf Lammen, 27-Feb-2014.)
⊢ ¬ ∀𝑦 ¬ 𝑥 = 𝑥
Theorem	ax4sp1 1469	A special case of ax-4 1443 without using ax-4 1443 or ax-17 1462. (Contributed by NM, 13-Jan-2011.)
⊢ (∀𝑦 ¬ 𝑥 = 𝑥 → ¬ 𝑥 = 𝑥)
1.3.5  Additional intuitionistic axioms
Axiom	ax-ial 1470	𝑥 is not free in ∀𝑥𝜑. One of the axioms of predicate logic. (Contributed by Mario Carneiro, 31-Jan-2015.)
⊢ (∀𝑥𝜑 → ∀𝑥∀𝑥𝜑)
Axiom	ax-i5r 1471	Axiom of quantifier collection. (Contributed by Mario Carneiro, 31-Jan-2015.)
⊢ ((∀𝑥𝜑 → ∀𝑥𝜓) → ∀𝑥(∀𝑥𝜑 → 𝜓))
1.3.6  Predicate calculus including ax-4, without distinct variables
Theorem	spi 1472	Inference rule reversing generalization. (Contributed by NM, 5-Aug-1993.)
⊢ ∀𝑥𝜑    ⇒   ⊢ 𝜑
Theorem	sps 1473	Generalization of antecedent. (Contributed by NM, 5-Aug-1993.)
⊢ (𝜑 → 𝜓)    ⇒   ⊢ (∀𝑥𝜑 → 𝜓)
Theorem	spsd 1474	Deduction generalizing antecedent. (Contributed by NM, 17-Aug-1994.)
⊢ (𝜑 → (𝜓 → 𝜒))    ⇒   ⊢ (𝜑 → (∀𝑥𝜓 → 𝜒))
Theorem	nfbidf 1475	An equality theorem for effectively not free. (Contributed by Mario Carneiro, 4-Oct-2016.)
⊢ Ⅎ𝑥𝜑    &   ⊢ (𝜑 → (𝜓 ↔ 𝜒))    ⇒   ⊢ (𝜑 → (Ⅎ𝑥𝜓 ↔ Ⅎ𝑥𝜒))
Theorem	hba1 1476	𝑥 is not free in ∀𝑥𝜑. Example in Appendix in [Megill] p. 450 (p. 19 of the preprint). Also Lemma 22 of [Monk2] p. 114. (Contributed by NM, 5-Aug-1993.)
⊢ (∀𝑥𝜑 → ∀𝑥∀𝑥𝜑)
Theorem	nfa1 1477	𝑥 is not free in ∀𝑥𝜑. (Contributed by Mario Carneiro, 11-Aug-2016.)
⊢ Ⅎ𝑥∀𝑥𝜑
Theorem	a5i 1478	Inference generalizing a consequent. (Contributed by NM, 5-Aug-1993.)
⊢ (∀𝑥𝜑 → 𝜓)    ⇒   ⊢ (∀𝑥𝜑 → ∀𝑥𝜓)
Theorem	nfnf1 1479	𝑥 is not free in Ⅎ𝑥𝜑. (Contributed by Mario Carneiro, 11-Aug-2016.)
⊢ Ⅎ𝑥Ⅎ𝑥𝜑
Theorem	hbim 1480	If 𝑥 is not free in 𝜑 and 𝜓, it is not free in (𝜑 → 𝜓). (Contributed by NM, 5-Aug-1993.) (Proof shortened by O'Cat, 3-Mar-2008.) (Revised by Mario Carneiro, 2-Feb-2015.)
⊢ (𝜑 → ∀𝑥𝜑)    &   ⊢ (𝜓 → ∀𝑥𝜓)    ⇒   ⊢ ((𝜑 → 𝜓) → ∀𝑥(𝜑 → 𝜓))
Theorem	hbor 1481	If 𝑥 is not free in 𝜑 and 𝜓, it is not free in (𝜑 ∨ 𝜓). (Contributed by NM, 5-Aug-1993.) (Revised by NM, 2-Feb-2015.)
⊢ (𝜑 → ∀𝑥𝜑)    &   ⊢ (𝜓 → ∀𝑥𝜓)    ⇒   ⊢ ((𝜑 ∨ 𝜓) → ∀𝑥(𝜑 ∨ 𝜓))
Theorem	hban 1482	If 𝑥 is not free in 𝜑 and 𝜓, it is not free in (𝜑 ∧ 𝜓). (Contributed by NM, 5-Aug-1993.) (Proof shortened by Mario Carneiro, 2-Feb-2015.)
⊢ (𝜑 → ∀𝑥𝜑)    &   ⊢ (𝜓 → ∀𝑥𝜓)    ⇒   ⊢ ((𝜑 ∧ 𝜓) → ∀𝑥(𝜑 ∧ 𝜓))
Theorem	hbbi 1483	If 𝑥 is not free in 𝜑 and 𝜓, it is not free in (𝜑 ↔ 𝜓). (Contributed by NM, 5-Aug-1993.)
⊢ (𝜑 → ∀𝑥𝜑)    &   ⊢ (𝜓 → ∀𝑥𝜓)    ⇒   ⊢ ((𝜑 ↔ 𝜓) → ∀𝑥(𝜑 ↔ 𝜓))
Theorem	hb3or 1484	If 𝑥 is not free in 𝜑, 𝜓, and 𝜒, it is not free in (𝜑 ∨ 𝜓 ∨ 𝜒). (Contributed by NM, 14-Sep-2003.)
⊢ (𝜑 → ∀𝑥𝜑)    &   ⊢ (𝜓 → ∀𝑥𝜓)    &   ⊢ (𝜒 → ∀𝑥𝜒)    ⇒   ⊢ ((𝜑 ∨ 𝜓 ∨ 𝜒) → ∀𝑥(𝜑 ∨ 𝜓 ∨ 𝜒))
Theorem	hb3an 1485	If 𝑥 is not free in 𝜑, 𝜓, and 𝜒, it is not free in (𝜑 ∧ 𝜓 ∧ 𝜒). (Contributed by NM, 14-Sep-2003.)
⊢ (𝜑 → ∀𝑥𝜑)    &   ⊢ (𝜓 → ∀𝑥𝜓)    &   ⊢ (𝜒 → ∀𝑥𝜒)    ⇒   ⊢ ((𝜑 ∧ 𝜓 ∧ 𝜒) → ∀𝑥(𝜑 ∧ 𝜓 ∧ 𝜒))
Theorem	hba2 1486	Lemma 24 of [Monk2] p. 114. (Contributed by NM, 29-May-2008.)
⊢ (∀𝑦∀𝑥𝜑 → ∀𝑥∀𝑦∀𝑥𝜑)
Theorem	hbia1 1487	Lemma 23 of [Monk2] p. 114. (Contributed by NM, 29-May-2008.)
⊢ ((∀𝑥𝜑 → ∀𝑥𝜓) → ∀𝑥(∀𝑥𝜑 → ∀𝑥𝜓))
Theorem	19.3h 1488	A wff may be quantified with a variable not free in it. Theorem 19.3 of [Margaris] p. 89. (Contributed by NM, 5-Aug-1993.) (Revised by NM, 21-May-2007.)
⊢ (𝜑 → ∀𝑥𝜑)    ⇒   ⊢ (∀𝑥𝜑 ↔ 𝜑)
Theorem	19.3 1489	A wff may be quantified with a variable not free in it. Theorem 19.3 of [Margaris] p. 89. (Contributed by NM, 5-Aug-1993.) (Revised by Mario Carneiro, 24-Sep-2016.)
⊢ Ⅎ𝑥𝜑    ⇒   ⊢ (∀𝑥𝜑 ↔ 𝜑)
Theorem	19.16 1490	Theorem 19.16 of [Margaris] p. 90. (Contributed by NM, 12-Mar-1993.)
⊢ Ⅎ𝑥𝜑    ⇒   ⊢ (∀𝑥(𝜑 ↔ 𝜓) → (𝜑 ↔ ∀𝑥𝜓))
Theorem	19.17 1491	Theorem 19.17 of [Margaris] p. 90. (Contributed by NM, 12-Mar-1993.)
⊢ Ⅎ𝑥𝜓    ⇒   ⊢ (∀𝑥(𝜑 ↔ 𝜓) → (∀𝑥𝜑 ↔ 𝜓))
Theorem	19.21h 1492	Theorem 19.21 of [Margaris] p. 90. The hypothesis can be thought of as "𝑥 is not free in 𝜑." New proofs should use 19.21 1518 instead. (Contributed by NM, 5-Aug-1993.) (New usage is discouraged.)
⊢ (𝜑 → ∀𝑥𝜑)    ⇒   ⊢ (∀𝑥(𝜑 → 𝜓) ↔ (𝜑 → ∀𝑥𝜓))
Theorem	19.21bi 1493	Inference from Theorem 19.21 of [Margaris] p. 90. (Contributed by NM, 5-Aug-1993.)
⊢ (𝜑 → ∀𝑥𝜓)    ⇒   ⊢ (𝜑 → 𝜓)
Theorem	19.21bbi 1494	Inference removing double quantifier. (Contributed by NM, 20-Apr-1994.)
⊢ (𝜑 → ∀𝑥∀𝑦𝜓)    ⇒   ⊢ (𝜑 → 𝜓)
Theorem	19.27h 1495	Theorem 19.27 of [Margaris] p. 90. (Contributed by NM, 5-Aug-1993.)
⊢ (𝜓 → ∀𝑥𝜓)    ⇒   ⊢ (∀𝑥(𝜑 ∧ 𝜓) ↔ (∀𝑥𝜑 ∧ 𝜓))
Theorem	19.27 1496	Theorem 19.27 of [Margaris] p. 90. (Contributed by NM, 5-Aug-1993.)
⊢ Ⅎ𝑥𝜓    ⇒   ⊢ (∀𝑥(𝜑 ∧ 𝜓) ↔ (∀𝑥𝜑 ∧ 𝜓))
Theorem	19.28h 1497	Theorem 19.28 of [Margaris] p. 90. (Contributed by NM, 5-Aug-1993.)
⊢ (𝜑 → ∀𝑥𝜑)    ⇒   ⊢ (∀𝑥(𝜑 ∧ 𝜓) ↔ (𝜑 ∧ ∀𝑥𝜓))
Theorem	19.28 1498	Theorem 19.28 of [Margaris] p. 90. (Contributed by NM, 5-Aug-1993.)
⊢ Ⅎ𝑥𝜑    ⇒   ⊢ (∀𝑥(𝜑 ∧ 𝜓) ↔ (𝜑 ∧ ∀𝑥𝜓))
Theorem	nfan1 1499	A closed form of nfan 1500. (Contributed by Mario Carneiro, 3-Oct-2016.)
⊢ Ⅎ𝑥𝜑    &   ⊢ (𝜑 → Ⅎ𝑥𝜓)    ⇒   ⊢ Ⅎ𝑥(𝜑 ∧ 𝜓)
Theorem	nfan 1500	If 𝑥 is not free in 𝜑 and 𝜓, it is not free in (𝜑 ∧ 𝜓). (Contributed by Mario Carneiro, 11-Aug-2016.) (Proof shortened by Wolf Lammen, 13-Jan-2018.)
⊢ Ⅎ𝑥𝜑    &   ⊢ Ⅎ𝑥𝜓    ⇒   ⊢ Ⅎ𝑥(𝜑 ∧ 𝜓)