P. 16 of Theorem List - Intuitionistic Logic Explorer

Theorem List for Intuitionistic Logic Explorer - 1501-1600   *Has distinct variable group(s)

Type	Label	Description
Statement
Theorem	mpgbir 1501	Modus ponens on biconditional combined with generalization. (Contributed by NM, 24-May-1994.) (Proof shortened by Stefan Allan, 28-Oct-2008.)
|- ( ph <-> A. x ps )   &    |- ps   =>    |- ph
Theorem	a7s 1502	Swap quantifiers in an antecedent. (Contributed by NM, 5-Aug-1993.)
|- ( A. x A. y ph -> ps )   =>    |- ( A. y A. x ph -> ps )
Theorem	alimi 1503	Inference quantifying both antecedent and consequent. (Contributed by NM, 5-Aug-1993.)
|- ( ph -> ps )   =>    |- ( A. x ph -> A. x ps )
Theorem	2alimi 1504	Inference doubly quantifying both antecedent and consequent. (Contributed by NM, 3-Feb-2005.)
|- ( ph -> ps )   =>    |- ( A. x A. y ph -> A. x A. y ps )
Theorem	alim 1505	Theorem 19.20 of [Margaris] p. 90. (Contributed by NM, 5-Aug-1993.) (Proof shortened by O'Cat, 30-Mar-2008.)
|- ( A. x ( ph -> ps ) -> ( A. x ph -> A. x ps ) )
Theorem	al2imi 1506	Inference quantifying antecedent, nested antecedent, and consequent. (Contributed by NM, 5-Aug-1993.)
|- ( ph -> ( ps -> ch ) )   =>    |- ( A. x ph -> ( A. x ps -> A. x ch ) )
Theorem	alanimi 1507	Variant of al2imi 1506 with conjunctive antecedent. (Contributed by Andrew Salmon, 8-Jun-2011.)
|- ( ( ph /\ ps ) -> ch )   =>    |- ( ( A. x ph /\ A. x ps ) -> A. x ch )
Syntax	wnf 1508	Extend wff definition to include the not-free predicate.
wff F/ x ph
Definition	df-nf 1509	Define the not-free predicate for wffs. This is read " x is not free in ph". Not-free means that the value of x cannot affect the value of ph, e.g., any occurrence of x in ph is effectively bound by a "for all" or something that expands to one (such as "there exists"). In particular, substitution for a variable not free in a wff does not affect its value (sbf 1825). An example of where this is used is stdpc5 1632. See nf2 1716 for an alternate definition which does not involve nested quantifiers on the same variable. Nonfreeness is a commonly used condition, so it is useful to have a notation for it. Surprisingly, there is no common formal notation for it, so here we devise one. Our definition lets us work with the notion of nonfreeness within the logic itself rather than as a metalogical side condition. To be precise, our definition really means "effectively not free", because it is slightly less restrictive than the usual textbook definition for "not free" (which considers syntactic freedom). For example, x is effectively not free in the expression x = x (even though x is syntactically free in it, so would be considered "free" in the usual textbook definition) because the value of x in the formula x = x does not affect the truth of that formula (and thus substitutions will not change the result), see nfequid 1750. (Contributed by Mario Carneiro, 11-Aug-2016.)
|- ( F/ x ph <-> A. x ( ph -> A. x ph ) )
Theorem	nfi 1510	Deduce that x is not free in ph from the definition. (Contributed by Mario Carneiro, 11-Aug-2016.)
|- ( ph -> A. x ph )   =>    |- F/ x ph
Theorem	hbth 1511	No variable is (effectively) free in a theorem. This and later "hypothesis-building" lemmas, with labels starting "hb...", allow us to construct proofs of formulas of the form |- ( ph -> A. x ph ) from smaller formulas of this form. These are useful for constructing hypotheses that state " x is (effectively) not free in ph". (Contributed by NM, 5-Aug-1993.)
|- ph   =>    |- ( ph -> A. x ph )
Theorem	nfth 1512	No variable is (effectively) free in a theorem. (Contributed by Mario Carneiro, 11-Aug-2016.)
|- ph   =>    |- F/ x ph
Theorem	nfnth 1513	No variable is (effectively) free in a non-theorem. (Contributed by Mario Carneiro, 6-Dec-2016.)
|- -. ph   =>    |- F/ x ph
Theorem	nftru 1514	The true constant has no free variables. (This can also be proven in one step with nfv 1576, but this proof does not use ax-17 1574.) (Contributed by Mario Carneiro, 6-Oct-2016.)
|- F/ x T.
Theorem	alimdh 1515	Deduction from Theorem 19.20 of [Margaris] p. 90. (Contributed by NM, 4-Jan-2002.)
|- ( ph -> A. x ph )   &    |- ( ph -> ( ps -> ch ) )   =>    |- ( ph -> ( A. x ps -> A. x ch ) )
Theorem	albi 1516	Theorem 19.15 of [Margaris] p. 90. (Contributed by NM, 5-Aug-1993.)
|- ( A. x ( ph <-> ps ) -> ( A. x ph <-> A. x ps ) )
Theorem	alrimih 1517	Inference from Theorem 19.21 of [Margaris] p. 90. (Contributed by NM, 5-Aug-1993.) (New usage is discouraged.)
|- ( ph -> A. x ph )   &    |- ( ph -> ps )   =>    |- ( ph -> A. x ps )
Theorem	albii 1518	Inference adding universal quantifier to both sides of an equivalence. (Contributed by NM, 7-Aug-1994.)
|- ( ph <-> ps )   =>    |- ( A. x ph <-> A. x ps )
Theorem	2albii 1519	Inference adding 2 universal quantifiers to both sides of an equivalence. (Contributed by NM, 9-Mar-1997.)
|- ( ph <-> ps )   =>    |- ( A. x A. y ph <-> A. x A. y ps )
Theorem	hbxfrbi 1520	A utility lemma to transfer a bound-variable hypothesis builder into a definition. (Contributed by Jonathan Ben-Naim, 3-Jun-2011.)
|- ( ph <-> ps )   &    |- ( ps -> A. x ps )   =>    |- ( ph -> A. x ph )
Theorem	nfbii 1521	Equality theorem for not-free. (Contributed by Mario Carneiro, 11-Aug-2016.)
|- ( ph <-> ps )   =>    |- ( F/ x ph <-> F/ x ps )
Theorem	nfxfr 1522	A utility lemma to transfer a bound-variable hypothesis builder into a definition. (Contributed by Mario Carneiro, 11-Aug-2016.)
|- ( ph <-> ps )   &    |- F/ x ps   =>    |- F/ x ph
Theorem	nfxfrd 1523	A utility lemma to transfer a bound-variable hypothesis builder into a definition. (Contributed by Mario Carneiro, 24-Sep-2016.)
|- ( ph <-> ps )   &    |- ( ch -> F/ x ps )   =>    |- ( ch -> F/ x ph )
Theorem	alcoms 1524	Swap quantifiers in an antecedent. (Contributed by NM, 11-May-1993.)
|- ( A. x A. y ph -> ps )   =>    |- ( A. y A. x ph -> ps )
Theorem	hbal 1525	If x is not free in ph, it is not free in A. y ph. (Contributed by NM, 5-Aug-1993.)
|- ( ph -> A. x ph )   =>    |- ( A. y ph -> A. x A. y ph )
Theorem	alcom 1526	Theorem 19.5 of [Margaris] p. 89. (Contributed by NM, 5-Aug-1993.)
|- ( A. x A. y ph <-> A. y A. x ph )
Theorem	alrimdh 1527	Deduction from Theorem 19.21 of [Margaris] p. 90. (Contributed by NM, 10-Feb-1997.) (Proof shortened by Andrew Salmon, 13-May-2011.)
|- ( ph -> A. x ph )   &    |- ( ps -> A. x ps )   &    |- ( ph -> ( ps -> ch ) )   =>    |- ( ph -> ( ps -> A. x ch ) )
Theorem	albidh 1528	Formula-building rule for universal quantifier (deduction form). (Contributed by NM, 5-Aug-1993.)
|- ( ph -> A. x ph )   &    |- ( ph -> ( ps <-> ch ) )   =>    |- ( ph -> ( A. x ps <-> A. x ch ) )
Theorem	19.26 1529	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.)
|- ( A. x ( ph /\ ps ) <-> ( A. x ph /\ A. x ps ) )
Theorem	19.26-2 1530	Theorem 19.26 of [Margaris] p. 90 with two quantifiers. (Contributed by NM, 3-Feb-2005.)
|- ( A. x A. y ( ph /\ ps ) <-> ( A. x A. y ph /\ A. x A. y ps ) )
Theorem	19.26-3an 1531	Theorem 19.26 of [Margaris] p. 90 with triple conjunction. (Contributed by NM, 13-Sep-2011.)
|- ( A. x ( ph /\ ps /\ ch ) <-> ( A. x ph /\ A. x ps /\ A. x ch ) )
Theorem	19.33 1532	Theorem 19.33 of [Margaris] p. 90. (Contributed by NM, 5-Aug-1993.)
|- ( ( A. x ph \/ A. x ps ) -> A. x ( ph \/ ps ) )
Theorem	alrot3 1533	Theorem *11.21 in [WhiteheadRussell] p. 160. (Contributed by Andrew Salmon, 24-May-2011.)
|- ( A. x A. y A. z ph <-> A. y A. z A. x ph )
Theorem	alrot4 1534	Rotate 4 universal quantifiers twice. (Contributed by NM, 2-Feb-2005.) (Proof shortened by Wolf Lammen, 28-Jun-2014.)
|- ( A. x A. y A. z A. w ph <-> A. z A. w A. x A. y ph )
Theorem	albiim 1535	Split a biconditional and distribute quantifier. (Contributed by NM, 18-Aug-1993.)
|- ( A. x ( ph <-> ps ) <-> ( A. x ( ph -> ps ) /\ A. x ( ps -> ph ) ) )
Theorem	2albiim 1536	Split a biconditional and distribute 2 quantifiers. (Contributed by NM, 3-Feb-2005.)
|- ( A. x A. y ( ph <-> ps ) <-> ( A. x A. y ( ph -> ps ) /\ A. x A. y ( ps -> ph ) ) )
Theorem	hband 1537	Deduction form of bound-variable hypothesis builder hban 1595. (Contributed by NM, 2-Jan-2002.)
|- ( ph -> ( ps -> A. x ps ) )   &    |- ( ph -> ( ch -> A. x ch ) )   =>    |- ( ph -> ( ( ps /\ ch ) -> A. x ( ps /\ ch ) ) )
Theorem	hb3and 1538	Deduction form of bound-variable hypothesis builder hb3an 1598. (Contributed by NM, 17-Feb-2013.)
|- ( ph -> ( ps -> A. x ps ) )   &    |- ( ph -> ( ch -> A. x ch ) )   &    |- ( ph -> ( th -> A. x th ) )   =>    |- ( ph -> ( ( ps /\ ch /\ th ) -> A. x ( ps /\ ch /\ th ) ) )
Theorem	hbald 1539	Deduction form of bound-variable hypothesis builder hbal 1525. (Contributed by NM, 2-Jan-2002.)
|- ( ph -> A. y ph )   &    |- ( ph -> ( ps -> A. x ps ) )   =>    |- ( ph -> ( A. y ps -> A. x A. y ps ) )
Syntax	wex 1540	Extend wff definition to include the existential quantifier ("there exists").
wff E. x ph
Axiom	ax-ie1 1541	x is bound in E. x ph. One of the axioms of predicate logic. (Contributed by Mario Carneiro, 31-Jan-2015.)
|- ( E. x ph -> A. x E. x ph )
Axiom	ax-ie2 1542	Define existential quantification. E. x ph means "there exists at least one set x such that ph is true". One of the axioms of predicate logic. (Contributed by Mario Carneiro, 31-Jan-2015.)
|- ( A. x ( ps -> A. x ps ) -> ( A. x ( ph -> ps ) <-> ( E. x ph -> ps ) ) )
Theorem	hbe1 1543	x is not free in E. x ph. (Contributed by NM, 5-Aug-1993.)
|- ( E. x ph -> A. x E. x ph )
Theorem	nfe1 1544	x is not free in E. x ph. (Contributed by Mario Carneiro, 11-Aug-2016.)
|- F/ x E. x ph
Theorem	19.23ht 1545	Closed form of Theorem 19.23 of [Margaris] p. 90. (Contributed by NM, 7-Nov-2005.) (Revised by Mario Carneiro, 1-Feb-2015.)
|- ( A. x ( ps -> A. x ps ) -> ( A. x ( ph -> ps ) <-> ( E. x ph -> ps ) ) )
Theorem	19.23h 1546	Theorem 19.23 of [Margaris] p. 90. (Contributed by NM, 5-Aug-1993.) (Revised by Mario Carneiro, 1-Feb-2015.)
|- ( ps -> A. x ps )   =>    |- ( A. x ( ph -> ps ) <-> ( E. x ph -> ps ) )
Theorem	alnex 1547	Theorem 19.7 of [Margaris] p. 89. To read this intuitionistically, think of it as "if ph can be refuted for all x, then it is not possible to find an x for which ph holds" (and likewise for the converse). Comparing this with dfexdc 1549 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.)
|- ( A. x -. ph <-> -. E. x ph )
Theorem	nex 1548	Generalization rule for negated wff. (Contributed by NM, 18-May-1994.)
|- -. ph   =>    |- -. E. x ph
Theorem	dfexdc 1549	Defining E. x ph given decidability. It is common in classical logic to define E. x ph as -. A. x -. ph but in intuitionistic logic without a decidability condition, that is only an implication not an equivalence, as seen at exalim 1550. (Contributed by Jim Kingdon, 15-Mar-2018.)
|- (DECID E. x ph -> ( E. x ph <-> -. A. x -. ph ) )
Theorem	exalim 1550	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 1549. (Contributed by Jim Kingdon, 29-Jul-2018.)
|- ( E. x ph -> -. A. x -. ph )
1.3.2  Equality predicate (continued) The equality predicate was introduced above in wceq 1397 for use by df-tru 1400. See the comments in that section. In this section, we continue with the first "real" use of it.
Theorem	weq 1551	Extend wff definition to include atomic formulas using the equality predicate. (Instead of introducing weq 1551 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 1397. This lets us avoid overloading the = connective, thus preventing ambiguity that would complicate certain Metamath parsers. However, logically weq 1551 is considered to be a primitive syntax, even though here it is artificially "derived" from wceq 1397. 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
Axiom	ax-8 1552	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 1757). Axiom scheme C8' in [Megill] p. 448 (p. 16 of the preprint). Also appears as Axiom C7 of [Monk2] p. 105. Axioms ax-8 1552 through ax-16 1862 are the axioms having to do with equality, substitution, and logical properties of our binary predicate e. (which later in set theory will mean "is a member of"). Note that all axioms except ax-16 1862 and ax-17 1574 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 1862 and ax-17 1574 only. (Contributed by NM, 5-Aug-1993.)
|- ( x = y -> ( x = z -> y = z ) )
Axiom	ax-10 1553	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 1764 ("o" for "old") and was replaced with this shorter ax-10 1553 in May 2008. The old axiom is proved from this one as Theorem ax10o 1763. Conversely, this axiom is proved from ax-10o 1764 as Theorem ax10 1765. (Contributed by NM, 5-Aug-1993.)
|- ( A. x x = y -> A. y y = x )
Axiom	ax-11 1554	Axiom of Variable Substitution. One of the 5 equality axioms of predicate calculus. The final consequent A. x ( x = y -> ph ) is a way of expressing " y substituted for x in wff ph " (cf. sb6 1935). 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 1875, ax11v2 1868 and ax-11o 1871. (Contributed by NM, 5-Aug-1993.)
|- ( x = y -> ( A. y ph -> A. x ( x = y -> ph ) ) )
Axiom	ax-i12 1555	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 ax12 1560 for compatibility with intuitionistic logic. (Contributed by Mario Carneiro, 31-Jan-2015.) Use its alias ax12or 1556 instead, for labeling consistency. (New usage is discouraged.)
|- ( A. z z = x \/ ( A. z z = y \/ A. z ( x = y -> A. z x = y ) ) )
Theorem	ax12or 1556	Alias for ax-i12 1555, to be used in place of it for labeling consistency. (Contributed by NM, 3-Feb-2015.)
|- ( A. z z = x \/ ( A. z z = y \/ A. z ( x = y -> A. z x = y ) ) )
Axiom	ax-bndl 1557	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 A. z z = x \/ -. A. z z = 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. A. z z = x holds if z and x are the same variable, likewise for z and y, and A. x A. z ( x = y -> A. z x = 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 1555 as can be seen at axi12 1562. Whether ax-bndl 1557 can be proved from the remaining axioms including ax-i12 1555 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.)
|- ( A. z z = x \/ ( A. z z = y \/ A. x A. z ( x = y -> A. z x = y ) ) )
Axiom	ax-4 1558	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 ph). (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 A. y x = 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 1497. Conditional forms of the converse are given by ax12 1560, ax-16 1862, and ax-17 1574. 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 1823. (Contributed by NM, 5-Aug-1993.)
|- ( A. x ph -> ph )
Theorem	sp 1559	Specialization. Another name for ax-4 1558. (Contributed by NM, 21-May-2008.)
|- ( A. x ph -> ph )
Theorem	ax12 1560	Rederive the original version of the axiom from ax-i12 1555. (Contributed by Mario Carneiro, 3-Feb-2015.)
|- ( -. A. z z = x -> ( -. A. z z = y -> ( x = y -> A. z x = y ) ) )
Theorem	hbequid 1561	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-8 1552 and ax-i12 1555 on top of (the FOL analogue of) modal logic KT. This shows that this can be proved without ax-i9 1578, even though Theorem equid 1749 cannot. A shorter proof using ax-i9 1578 is obtainable from equid 1749 and hbth 1511. (Contributed by NM, 13-Jan-2011.) (Proof shortened by Wolf Lammen, 23-Mar-2014.)
|- ( x = x -> A. y x = x )
Theorem	axi12 1562	Proof that ax-i12 1555 follows from ax-bndl 1557. So that we can track which theorems rely on ax-bndl 1557, proofs should reference ax12or 1556 rather than this theorem. (Contributed by Jim Kingdon, 17-Aug-2018.) (New usage is discouraged). (Proof modification is discouraged.)
|- ( A. z z = x \/ ( A. z z = y \/ A. z ( x = y -> A. z x = y ) ) )
Theorem	alequcom 1563	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.)
|- ( A. x x = y -> A. y y = x )
Theorem	alequcoms 1564	A commutation rule for identical variable specifiers. (Contributed by NM, 5-Aug-1993.)
|- ( A. x x = y -> ph )   =>    |- ( A. y y = x -> ph )
Theorem	nalequcoms 1565	A commutation rule for distinct variable specifiers. (Contributed by NM, 2-Jan-2002.) (Revised by Mario Carneiro, 2-Feb-2015.)
|- ( -. A. x x = y -> ph )   =>    |- ( -. A. y y = x -> ph )
Theorem	nfr 1566	Consequence of the definition of not-free. (Contributed by Mario Carneiro, 26-Sep-2016.)
|- ( F/ x ph -> ( ph -> A. x ph ) )
Theorem	nfri 1567	Consequence of the definition of not-free. (Contributed by Mario Carneiro, 11-Aug-2016.)
|- F/ x ph   =>    |- ( ph -> A. x ph )
Theorem	nfrd 1568	Consequence of the definition of not-free in a context. (Contributed by Mario Carneiro, 11-Aug-2016.)
|- ( ph -> F/ x ps )   =>    |- ( ph -> ( ps -> A. x ps ) )
Theorem	alimd 1569	Deduction from Theorem 19.20 of [Margaris] p. 90. (Contributed by Mario Carneiro, 24-Sep-2016.)
|- F/ x ph   &    |- ( ph -> ( ps -> ch ) )   =>    |- ( ph -> ( A. x ps -> A. x ch ) )
Theorem	alrimi 1570	Inference from Theorem 19.21 of [Margaris] p. 90. (Contributed by Mario Carneiro, 24-Sep-2016.)
|- F/ x ph   &    |- ( ph -> ps )   =>    |- ( ph -> A. x ps )
Theorem	nfd 1571	Deduce that x is not free in ps in a context. (Contributed by Mario Carneiro, 24-Sep-2016.)
|- F/ x ph   &    |- ( ph -> ( ps -> A. x ps ) )   =>    |- ( ph -> F/ x ps )
Theorem	nfdh 1572	Deduce that x is not free in ps in a context. (Contributed by Mario Carneiro, 24-Sep-2016.)
|- ( ph -> A. x ph )   &    |- ( ph -> ( ps -> A. x ps ) )   =>    |- ( ph -> F/ x ps )
Theorem	nfrimi 1573	Moving an antecedent outside F/. (Contributed by Jim Kingdon, 23-Mar-2018.)
|- F/ x ph   &    |- F/ x ( ph -> ps )   =>    |- ( ph -> F/ x ps )
1.3.3  Axiom ax-17 - first use of the $d distinct variable statement
Axiom	ax-17 1574*	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.)
|- ( ph -> A. x ph )
Theorem	a17d 1575*	ax-17 1574 with antecedent. (Contributed by NM, 1-Mar-2013.)
|- ( ph -> ( ps -> A. x ps ) )
Theorem	nfv 1576*	If x is not present in ph, then x is not free in ph. (Contributed by Mario Carneiro, 11-Aug-2016.)
|- F/ x ph
Theorem	nfvd 1577*	nfv 1576 with antecedent. Useful in proofs of deduction versions of bound-variable hypothesis builders such as nfimd 1633. (Contributed by Mario Carneiro, 6-Oct-2016.)
|- ( ph -> F/ x ps )
1.3.4  Introduce Axiom of Existence
Axiom	ax-i9 1578	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 1558 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 1744, which has additional remarks. (Contributed by Mario Carneiro, 31-Jan-2015.)
|- E. x x = y
Theorem	ax-9 1579	Derive ax-9 1579 from ax-i9 1578, the modified version for intuitionistic logic. Although ax-9 1579 does hold intuistionistically, in intuitionistic logic it is weaker than ax-i9 1578. (Contributed by NM, 3-Feb-2015.)
|- -. A. x -. x = y
Theorem	equidqe 1580	equid 1749 with some quantification and negation without using ax-4 1558 or ax-17 1574. (Contributed by NM, 13-Jan-2011.) (Proof shortened by Wolf Lammen, 27-Feb-2014.)
|- -. A. y -. x = x
Theorem	ax4sp1 1581	A special case of ax-4 1558 without using ax-4 1558 or ax-17 1574. (Contributed by NM, 13-Jan-2011.)
|- ( A. y -. x = x -> -. x = x )
1.3.5  Additional intuitionistic axioms
Axiom	ax-ial 1582	x is not free in A. x ph. One of the axioms of predicate logic. (Contributed by Mario Carneiro, 31-Jan-2015.)
|- ( A. x ph -> A. x A. x ph )
Axiom	ax-i5r 1583	Axiom of quantifier collection. (Contributed by Mario Carneiro, 31-Jan-2015.)
|- ( ( A. x ph -> A. x ps ) -> A. x ( A. x ph -> ps ) )
1.3.6  Predicate calculus including ax-4, without distinct variables
Theorem	spi 1584	Inference reversing generalization (specialization). (Contributed by NM, 5-Aug-1993.)
|- A. x ph   =>    |- ph
Theorem	sps 1585	Generalization of antecedent. (Contributed by NM, 5-Aug-1993.)
|- ( ph -> ps )   =>    |- ( A. x ph -> ps )
Theorem	spsd 1586	Deduction generalizing antecedent. (Contributed by NM, 17-Aug-1994.)
|- ( ph -> ( ps -> ch ) )   =>    |- ( ph -> ( A. x ps -> ch ) )
Theorem	nfbidf 1587	An equality theorem for effectively not free. (Contributed by Mario Carneiro, 4-Oct-2016.)
|- F/ x ph   &    |- ( ph -> ( ps <-> ch ) )   =>    |- ( ph -> ( F/ x ps <-> F/ x ch ) )
Theorem	hba1 1588	x is not free in A. x ph. 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.)
|- ( A. x ph -> A. x A. x ph )
Theorem	nfa1 1589	x is not free in A. x ph. (Contributed by Mario Carneiro, 11-Aug-2016.)
|- F/ x A. x ph
Theorem	axc4i 1590	Inference version of 19.21 1631. (Contributed by NM, 3-Jan-1993.)
|- ( A. x ph -> ps )   =>    |- ( A. x ph -> A. x ps )
Theorem	a5i 1591	Inference generalizing a consequent. (Contributed by NM, 5-Aug-1993.)
|- ( A. x ph -> ps )   =>    |- ( A. x ph -> A. x ps )
Theorem	nfnf1 1592	x is not free in F/ x ph. (Contributed by Mario Carneiro, 11-Aug-2016.)
|- F/ x F/ x ph
Theorem	hbim 1593	If x is not free in ph and ps, it is not free in ( ph -> ps ). (Contributed by NM, 5-Aug-1993.) (Proof shortened by O'Cat, 3-Mar-2008.) (Revised by Mario Carneiro, 2-Feb-2015.)
|- ( ph -> A. x ph )   &    |- ( ps -> A. x ps )   =>    |- ( ( ph -> ps ) -> A. x ( ph -> ps ) )
Theorem	hbor 1594	If x is not free in ph and ps, it is not free in ( ph \/ ps ). (Contributed by NM, 5-Aug-1993.) (Revised by NM, 2-Feb-2015.)
|- ( ph -> A. x ph )   &    |- ( ps -> A. x ps )   =>    |- ( ( ph \/ ps ) -> A. x ( ph \/ ps ) )
Theorem	hban 1595	If x is not free in ph and ps, it is not free in ( ph /\ ps ). (Contributed by NM, 5-Aug-1993.) (Proof shortened by Mario Carneiro, 2-Feb-2015.)
|- ( ph -> A. x ph )   &    |- ( ps -> A. x ps )   =>    |- ( ( ph /\ ps ) -> A. x ( ph /\ ps ) )
Theorem	hbbi 1596	If x is not free in ph and ps, it is not free in ( ph <-> ps ). (Contributed by NM, 5-Aug-1993.)
|- ( ph -> A. x ph )   &    |- ( ps -> A. x ps )   =>    |- ( ( ph <-> ps ) -> A. x ( ph <-> ps ) )
Theorem	hb3or 1597	If x is not free in ph, ps, and ch, it is not free in ( ph \/ ps \/ ch ). (Contributed by NM, 14-Sep-2003.)
|- ( ph -> A. x ph )   &    |- ( ps -> A. x ps )   &    |- ( ch -> A. x ch )   =>    |- ( ( ph \/ ps \/ ch ) -> A. x ( ph \/ ps \/ ch ) )
Theorem	hb3an 1598	If x is not free in ph, ps, and ch, it is not free in ( ph /\ ps /\ ch ). (Contributed by NM, 14-Sep-2003.)
|- ( ph -> A. x ph )   &    |- ( ps -> A. x ps )   &    |- ( ch -> A. x ch )   =>    |- ( ( ph /\ ps /\ ch ) -> A. x ( ph /\ ps /\ ch ) )
Theorem	hba2 1599	Lemma 24 of [Monk2] p. 114. (Contributed by NM, 29-May-2008.)
|- ( A. y A. x ph -> A. x A. y A. x ph )
Theorem	hbia1 1600	Lemma 23 of [Monk2] p. 114. (Contributed by NM, 29-May-2008.)
|- ( ( A. x ph -> A. x ps ) -> A. x ( A. x ph -> A. x ps ) )