P. 17 of Theorem List - Metamath Proof Explorer

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Theorem List for Metamath Proof Explorer - 1601-1700   *Has distinct variable group(s)

Type	Label	Description
Statement
Theorem	19.26-2 1601	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 1602	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	19.29 1603	Theorem 19.29 of [Margaris] p. 90. (Contributed by NM, 5-Aug-1993.) (Proof shortened by Andrew Salmon, 13-May-2011.)
|- ( ( A. x ph /\ E. x ps ) -> E. x ( ph /\ ps ) )
Theorem	19.29r 1604	Variation of Theorem 19.29 of [Margaris] p. 90. (Contributed by NM, 18-Aug-1993.)
|- ( ( E. x ph /\ A. x ps ) -> E. x ( ph /\ ps ) )
Theorem	19.29r2 1605	Variation of Theorem 19.29 of [Margaris] p. 90 with double quantification. (Contributed by NM, 3-Feb-2005.)
|- ( ( E. x E. y ph /\ A. x A. y ps ) -> E. x E. y ( ph /\ ps ) )
Theorem	19.29x 1606	Variation of Theorem 19.29 of [Margaris] p. 90 with mixed quantification. (Contributed by NM, 11-Feb-2005.)
|- ( ( E. x A. y ph /\ A. x E. y ps ) -> E. x E. y ( ph /\ ps ) )
Theorem	19.35 1607	Theorem 19.35 of [Margaris] p. 90. This theorem is useful for moving an implication (in the form of the right-hand side) into the scope of a single existential quantifier. (Contributed by NM, 5-Aug-1993.) (Proof shortened by Wolf Lammen, 27-Jun-2014.)
|- ( E. x ( ph -> ps ) <-> ( A. x ph -> E. x ps ) )
Theorem	19.35i 1608	Inference from Theorem 19.35 of [Margaris] p. 90. (Contributed by NM, 5-Aug-1993.)
|- E. x ( ph -> ps )   =>    |- ( A. x ph -> E. x ps )
Theorem	19.35ri 1609	Inference from Theorem 19.35 of [Margaris] p. 90. (Contributed by NM, 5-Aug-1993.)
|- ( A. x ph -> E. x ps )   =>    |- E. x ( ph -> ps )
Theorem	19.25 1610	Theorem 19.25 of [Margaris] p. 90. (Contributed by NM, 5-Aug-1993.)
|- ( A. y E. x ( ph -> ps ) -> ( E. y A. x ph -> E. y E. x ps ) )
Theorem	19.30 1611	Theorem 19.30 of [Margaris] p. 90. (Contributed by NM, 5-Aug-1993.) (Proof shortened by Andrew Salmon, 25-May-2011.)
|- ( A. x ( ph \/ ps ) -> ( A. x ph \/ E. x ps ) )
Theorem	19.43 1612	Theorem 19.43 of [Margaris] p. 90. (Contributed by NM, 5-Aug-1993.) (Proof shortened by Wolf Lammen, 27-Jun-2014.)
|- ( E. x ( ph \/ ps ) <-> ( E. x ph \/ E. x ps ) )
Theorem	19.43OLD 1613	Obsolete proof of 19.43 1612 as of 3-May-2017. Leave this in for the example on the mmrecent.html page. (Contributed by NM, 5-Aug-1993.) (Proof modification is discouraged.) (New usage is discouraged.)
|- ( E. x ( ph \/ ps ) <-> ( E. x ph \/ E. x ps ) )
Theorem	19.33 1614	Theorem 19.33 of [Margaris] p. 90. (Contributed by NM, 5-Aug-1993.)
|- ( ( A. x ph \/ A. x ps ) -> A. x ( ph \/ ps ) )
Theorem	19.33b 1615	The antecedent provides a condition implying the converse of 19.33 1614. Compare Theorem 19.33 of [Margaris] p. 90. (Contributed by NM, 27-Mar-2004.) (Proof shortened by Andrew Salmon, 25-May-2011.) (Proof shortened by Wolf Lammen, 5-Jul-2014.)
|- ( -. ( E. x ph /\ E. x ps ) -> ( A. x ( ph \/ ps ) <-> ( A. x ph \/ A. x ps ) ) )
Theorem	19.40 1616	Theorem 19.40 of [Margaris] p. 90. (Contributed by NM, 5-Aug-1993.)
|- ( E. x ( ph /\ ps ) -> ( E. x ph /\ E. x ps ) )
Theorem	19.40-2 1617	Theorem *11.42 in [WhiteheadRussell] p. 163. Theorem 19.40 of [Margaris] p. 90 with 2 quantifiers. (Contributed by Andrew Salmon, 24-May-2011.)
|- ( E. x E. y ( ph /\ ps ) -> ( E. x E. y ph /\ E. x E. y ps ) )
Theorem	albiim 1618	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 1619	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	exintrbi 1620	Add/remove a conjunct in the scope of an existential quantifier. (Contributed by Raph Levien, 3-Jul-2006.)
|- ( A. x ( ph -> ps ) -> ( E. x ph <-> E. x ( ph /\ ps ) ) )
Theorem	exintr 1621	Introduce a conjunct in the scope of an existential quantifier. (Contributed by NM, 11-Aug-1993.)
|- ( A. x ( ph -> ps ) -> ( E. x ph -> E. x ( ph /\ ps ) ) )
Theorem	alsyl 1622	Theorem *10.3 in [WhiteheadRussell] p. 150. (Contributed by Andrew Salmon, 8-Jun-2011.)
|- ( ( A. x ( ph -> ps ) /\ A. x ( ps -> ch ) ) -> A. x ( ph -> ch ) )
1.4.4  Axiom scheme ax-17 (Distinctness) - first use of $d
Axiom	ax-17 1623*	Axiom of Distinctness. This axiom quantifies 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. (See comments in ax17o 2207 about the logical redundancy of ax-17 1623 in the presence of our obsolete axioms.) This axiom essentially says that if x does not occur in ph, i.e. ph does not depend on x in any way, then we can add the quantifier A. x to ph with no further assumptions. By sp 1759, we can also remove the quantifier (unconditionally). (Contributed by NM, 5-Aug-1993.)
|- ( ph -> A. x ph )
Theorem	a17d 1624*	ax-17 1623 with antecedent. Useful in proofs of deduction versions of bound-variable hypothesis builders. (Contributed by NM, 1-Mar-2013.)
|- ( ph -> ( ps -> A. x ps ) )
Theorem	ax17e 1625*	A rephrasing of ax-17 1623 using the existential quantifier. (Contributed by Wolf Lammen, 4-Dec-2017.)
|- ( E. x ph -> ph )
Theorem	nfv 1626*	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 1627*	nfv 1626 with antecedent. Useful in proofs of deduction versions of bound-variable hypothesis builders such as nfimd 1823. (Contributed by Mario Carneiro, 6-Oct-2016.)
|- ( ph -> F/ x ps )
Theorem	alimdv 1628*	Deduction from Theorem 19.20 of [Margaris] p. 90. (Contributed by NM, 3-Apr-1994.)
|- ( ph -> ( ps -> ch ) )   =>    |- ( ph -> ( A. x ps -> A. x ch ) )
Theorem	eximdv 1629*	Deduction from Theorem 19.22 of [Margaris] p. 90. (Contributed by NM, 27-Apr-1994.)
|- ( ph -> ( ps -> ch ) )   =>    |- ( ph -> ( E. x ps -> E. x ch ) )
Theorem	2alimdv 1630*	Deduction from Theorem 19.22 of [Margaris] p. 90. (Contributed by NM, 27-Apr-2004.)
|- ( ph -> ( ps -> ch ) )   =>    |- ( ph -> ( A. x A. y ps -> A. x A. y ch ) )
Theorem	2eximdv 1631*	Deduction from Theorem 19.22 of [Margaris] p. 90. (Contributed by NM, 3-Aug-1995.)
|- ( ph -> ( ps -> ch ) )   =>    |- ( ph -> ( E. x E. y ps -> E. x E. y ch ) )
Theorem	albidv 1632*	Formula-building rule for universal quantifier (deduction rule). (Contributed by NM, 5-Aug-1993.)
|- ( ph -> ( ps <-> ch ) )   =>    |- ( ph -> ( A. x ps <-> A. x ch ) )
Theorem	exbidv 1633*	Formula-building rule for existential quantifier (deduction rule). (Contributed by NM, 5-Aug-1993.)
|- ( ph -> ( ps <-> ch ) )   =>    |- ( ph -> ( E. x ps <-> E. x ch ) )
Theorem	2albidv 1634*	Formula-building rule for 2 universal quantifiers (deduction rule). (Contributed by NM, 4-Mar-1997.)
|- ( ph -> ( ps <-> ch ) )   =>    |- ( ph -> ( A. x A. y ps <-> A. x A. y ch ) )
Theorem	2exbidv 1635*	Formula-building rule for 2 existential quantifiers (deduction rule). (Contributed by NM, 1-May-1995.)
|- ( ph -> ( ps <-> ch ) )   =>    |- ( ph -> ( E. x E. y ps <-> E. x E. y ch ) )
Theorem	3exbidv 1636*	Formula-building rule for 3 existential quantifiers (deduction rule). (Contributed by NM, 1-May-1995.)
|- ( ph -> ( ps <-> ch ) )   =>    |- ( ph -> ( E. x E. y E. z ps <-> E. x E. y E. z ch ) )
Theorem	4exbidv 1637*	Formula-building rule for 4 existential quantifiers (deduction rule). (Contributed by NM, 3-Aug-1995.)
|- ( ph -> ( ps <-> ch ) )   =>    |- ( ph -> ( E. x E. y E. z E. w ps <-> E. x E. y E. z E. w ch ) )
Theorem	alrimiv 1638*	Inference from Theorem 19.21 of [Margaris] p. 90. (Contributed by NM, 5-Aug-1993.)
|- ( ph -> ps )   =>    |- ( ph -> A. x ps )
Theorem	alrimivv 1639*	Inference from Theorem 19.21 of [Margaris] p. 90. (Contributed by NM, 31-Jul-1995.)
|- ( ph -> ps )   =>    |- ( ph -> A. x A. y ps )
Theorem	alrimdv 1640*	Deduction from Theorem 19.21 of [Margaris] p. 90. (Contributed by NM, 10-Feb-1997.)
|- ( ph -> ( ps -> ch ) )   =>    |- ( ph -> ( ps -> A. x ch ) )
Theorem	exlimiv 1641*	Inference from Theorem 19.23 of [Margaris] p. 90. This inference, along with our many variants such as rexlimdv 2789, is used to implement a metatheorem called "Rule C" that is given in many logic textbooks. See, for example, Rule C in [Mendelson] p. 81, Rule C in [Margaris] p. 40, or Rule C in Hirst and Hirst's A Primer for Logic and Proof p. 59 (PDF p. 65) at http://www.mathsci.appstate.edu/~hirstjl/primer/hirst.pdf. In informal proofs, the statement "Let C be an element such that..." almost always means an implicit application of Rule C. In essence, Rule C states that if we can prove that some element x exists satisfying a wff, i.e. E. x ph ( x ) where ph ( x ) has x free, then we can use ph ( C ) as a hypothesis for the proof where C is a new (ficticious) constant not appearing previously in the proof, nor in any axioms used, nor in the theorem to be proved. The purpose of Rule C is to get rid of the existential quantifier. We cannot do this in Metamath directly. Instead, we use the original ph (containing x) as an antecedent for the main part of the proof. We eventually arrive at ( ph -> ps ) where ps is the theorem to be proved and does not contain x. Then we apply exlimiv 1641 to arrive at ( E. x ph -> ps ). Finally, we separately prove E. x ph and detach it with modus ponens ax-mp 8 to arrive at the final theorem ps. (Contributed by NM, 5-Aug-1993.) (Revised by Wolf Lammen to remove dependency on ax-9 and ax-8, 4-Dec-2017.)
|- ( ph -> ps )   =>    |- ( E. x ph -> ps )
Theorem	exlimivv 1642*	Inference from Theorem 19.23 of [Margaris] p. 90. (Contributed by NM, 1-Aug-1995.)
|- ( ph -> ps )   =>    |- ( E. x E. y ph -> ps )
Theorem	exlimdv 1643*	Deduction from Theorem 19.23 of [Margaris] p. 90. Revised to remove dependency on ax-9 1662 and ax-8 1683. (Contributed by NM, 27-Apr-1994.) (Revised by Wolf Lammen, 4-Dec-2017.)
|- ( ph -> ( ps -> ch ) )   =>    |- ( ph -> ( E. x ps -> ch ) )
Theorem	exlimdvv 1644*	Deduction from Theorem 19.23 of [Margaris] p. 90. (Contributed by NM, 31-Jul-1995.)
|- ( ph -> ( ps -> ch ) )   =>    |- ( ph -> ( E. x E. y ps -> ch ) )
Theorem	exlimddv 1645*	Existential elimination rule of natural deduction. (Contributed by Mario Carneiro, 15-Jun-2016.)
|- ( ph -> E. x ps )   &    |- ( ( ph /\ ps ) -> ch )   =>    |- ( ph -> ch )
Theorem	nfdv 1646*	Apply the definition of not-free in a context. (Contributed by Mario Carneiro, 11-Aug-2016.)
|- ( ph -> ( ps -> A. x ps ) )   =>    |- ( ph -> F/ x ps )
Theorem	2ax17 1647*	Quantification of two variables over a formula in which they do not occur. (Contributed by Alan Sare, 12-Apr-2011.)
|- ( ph -> A. x A. y ph )
1.4.5  Equality predicate; define substitution
Syntax	cv 1648	This syntax construction states that a variable x, which has been declared to be a set variable by $f statement vx, is also a class expression. This can be justified informally as follows. We know that the class builder { y | y e. x } is a class by cab 2390. Since (when y is distinct from x) we have x = { y | y e. x } by cvjust 2399, we can argue that the syntax " class x " can be viewed as an abbreviation for " class { y | y e. x }". See the discussion under the definition of class in [Jech] p. 4 showing that "Every set can be considered to be a class." While it is tempting and perhaps occasionally useful to view cv 1648 as a "type conversion" from a set variable to a class variable, keep in mind that cv 1648 is intrinsically no different from any other class-building syntax such as cab 2390, cun 3278, or c0 3588. For a general discussion of the theory of classes and the role of cv 1648, see http://us.metamath.org/mpeuni/mmset.html#class. (The description above applies to set theory, not predicate calculus. The purpose of introducing class x here, and not in set theory where it belongs, is to allow us to express i.e. "prove" the weq 1650 of predicate calculus from the wceq 1649 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.)
class x
Syntax	wceq 1649	Extend wff definition to include class equality. For a general discussion of the theory of classes, see http://us.metamath.org/mpeuni/mmset.html#class. (The purpose of introducing wff A = B here, and not in set theory where it belongs, is to allow us to express i.e. "prove" the weq 1650 of predicate calculus in terms of the wceq 1649 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. For example, some parsers - although not the Metamath program - stumble on the fact that the = in x = y could be the = of either weq 1650 or wceq 1649, although mathematically it makes no difference. The class variables A and B are introduced temporarily for the purpose of this definition but otherwise not used in predicate calculus. See df-cleq 2397 for more information on the set theory usage of wceq 1649.)
wff A = B
Theorem	weq 1650	Extend wff definition to include atomic formulas using the equality predicate. (Instead of introducing weq 1650 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 1649. This lets us avoid overloading the = connective, thus preventing ambiguity that would complicate certain Metamath parsers. However, logically weq 1650 is considered to be a primitive syntax, even though here it is artificially "derived" from wceq 1649. 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
Theorem	equs3 1651	Lemma used in proofs of substitution properties. (Contributed by NM, 5-Aug-1993.)
|- ( E. x ( x = y /\ ph ) <-> -. A. x ( x = y -> -. ph ) )
Theorem	speimfw 1652	Specialization, with additional weakening to allow bundling of x and y. Uses only Tarski's FOL axiom schemes. (Contributed by NM, 23-Apr-2017.) (Proof shortened by Wolf Lammen, 5-Aug-2017.)
|- ( x = y -> ( ph -> ps ) )   =>    |- ( -. A. x -. x = y -> ( A. x ph -> E. x ps ) )
Theorem	spimfw 1653	Specialization, with additional weakening to allow bundling of x and y. Uses only Tarski's FOL axiom schemes. (Contributed by NM, 23-Apr-1017.) (Proof shortened by Wolf Lammen, 7-Aug-2017.)
|- ( -. ps -> A. x -. ps )   &    |- ( x = y -> ( ph -> ps ) )   =>    |- ( -. A. x -. x = y -> ( A. x ph -> ps ) )
Theorem	ax11i 1654	Inference that has ax-11 1757 (without A. y) as its conclusion. Uses only Tarski's FOL axiom schemes. The hypotheses may be eliminable without one or more of these axioms in special cases. Proof similar to Lemma 16 of [Tarski] p. 70. (Contributed by NM, 20-May-2008.)
|- ( x = y -> ( ph <-> ps ) )   &    |- ( ps -> A. x ps )   =>    |- ( x = y -> ( ph -> A. x ( x = y -> ph ) ) )
Syntax	wsb 1655	Extend wff definition to include proper substitution (read "the wff that results when y is properly substituted for x in wff ph"). (Contributed by NM, 24-Jan-2006.)
wff [ y / x ] ph
Definition	df-sb 1656	Define proper substitution. Remark 9.1 in [Megill] p. 447 (p. 15 of the preprint). For our notation, we use [ y / x ] ph to mean "the wff that results from the proper substitution of y for x in the wff ph." We can also use [ y / x ] ph in place of the "free for" side condition used in traditional predicate calculus; see, for example, stdpc4 2073. Our notation was introduced in Haskell B. Curry's Foundations of Mathematical Logic (1977), p. 316 and is frequently used in textbooks of lambda calculus and combinatory logic. This notation improves the common but ambiguous notation, " ph ( y ) is the wff that results when y is properly substituted for x in ph ( x )." For example, if the original ph ( x ) is x = y, then ph ( y ) is y = y, from which we obtain that ph ( x ) is x = x. So what exactly does ph ( x ) mean? Curry's notation solves this problem. In most books, proper substitution has a somewhat complicated recursive definition with multiple cases based on the occurrences of free and bound variables in the wff. Instead, we use a single formula that is exactly equivalent and gives us a direct definition. We later prove that our definition has the properties we expect of proper substitution (see theorems sbequ 2109, sbcom2 2163 and sbid2v 2173). Note that our definition is valid even when x and y are replaced with the same variable, as sbid 1943 shows. We achieve this by having x free in the first conjunct and bound in the second. We can also achieve this by using a dummy variable, as the alternate definition dfsb7 2171 shows (which some logicians may prefer because it doesn't mix free and bound variables). Another version that mixes free and bound variables is dfsb3 2105. When x and y are distinct, we can express proper substitution with the simpler expressions of sb5 2149 and sb6 2148. There are no restrictions on any of the variables, including what variables may occur in wff ph. (Contributed by NM, 5-Aug-1993.)
|- ( [ y / x ] ph <-> ( ( x = y -> ph ) /\ E. x ( x = y /\ ph ) ) )
Theorem	sbequ2 1657	An equality theorem for substitution. (Contributed by NM, 5-Aug-1993.) (Proof shortened by Wolf Lammen, 25-Feb-2018.)
|- ( x = y -> ( [ y / x ] ph -> ph ) )
Theorem	sbequ2OLD 1658	Obsolete proof of sbequ2 1657 as of 25-Feb-2018. (Contributed by NM, 5-Aug-1993.) (New usage is discouraged.) (Proof modification is discouraged.)
|- ( x = y -> ( [ y / x ] ph -> ph ) )
Theorem	sb1 1659	One direction of a simplified definition of substitution. (Contributed by NM, 5-Aug-1993.)
|- ( [ y / x ] ph -> E. x ( x = y /\ ph ) )
Theorem	sbimi 1660	Infer substitution into antecedent and consequent of an implication. (Contributed by NM, 25-Jun-1998.)
|- ( ph -> ps )   =>    |- ( [ y / x ] ph -> [ y / x ] ps )
Theorem	sbbii 1661	Infer substitution into both sides of a logical equivalence. (Contributed by NM, 5-Aug-1993.)
|- ( ph <-> ps )   =>    |- ( [ y / x ] ph <-> [ y / x ] ps )
1.4.6  Axiom scheme ax-9 (Existence)
Axiom	ax-9 1662	Axiom of Existence. One of the equality and substitution axioms of predicate calculus with equality. This axiom tells us is that at least one thing exists. 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.) It is equivalent to axiom scheme C10' in [Megill] p. 448 (p. 16 of the preprint); the equivalence is established by ax9o 1950 and ax9from9o 2198. A more convenient form of this axiom is a9e 1948, which has additional remarks. Raph Levien proved the independence of this axiom from the other logical axioms on 12-Apr-2005. See item 16 at http://us.metamath.org/award2003.html. ax-9 1662 can be proved from the weaker version ax9v 1663 requiring that the variables be distinct; see theorem ax9 1949. ax-9 1662 can also be proved from the Axiom of Separation (in the form that we use that axiom, where free variables are not universally quantified). See theorem ax9vsep 4294. Except by ax9v 1663, this axiom should not be referenced directly. Instead, use theorem ax9 1949. (Contributed by NM, 5-Aug-1993.) (New usage is discouraged.)
|- -. A. x -. x = y
Theorem	ax9v 1663*	Axiom B7 of [Tarski] p. 75, which requires that x and y be distinct. This trivial proof is intended merely to weaken axiom ax-9 1662 by adding a distinct variable restriction. From here on, ax-9 1662 should not be referenced directly by any other proof, so that theorem ax9 1949 will show that we can recover ax-9 1662 from this weaker version if it were an axiom (as it is in the case of Tarski). Note: Introducing x y as a distinct variable group "out of the blue" with no apparent justification has puzzled some people, but it is perfectly sound. All we are doing is adding an additional redundant requirement, no different from adding a redundant logical hypothesis, that results in a weakening of the theorem. This means that any future theorem that references ax9v 1663 must have a $d specified for the two variables that get substituted for x and y. The $d does not propagate "backwards" i.e. it does not impose a requirement on ax-9 1662. When possible, use of this theorem rather than ax9 1949 is preferred since its derivation from axioms is much shorter. (Contributed by NM, 7-Aug-2015.)
|- -. A. x -. x = y
Theorem	a9ev 1664*	At least one individual exists. Weaker version of a9e 1948. When possible, use of this theorem rather than a9e 1948 is preferred since its derivation from axioms is much shorter. (Contributed by NM, 3-Aug-2017.)
|- E. x x = y
Theorem	exiftru 1665	A companion rule to ax-gen, valid only if an individual exists. Unlike ax-9 1662, it does not require equality on its interface. Some fundamental theorems of predicate logic can be proven from ax-gen 1552, ax-5 1563 and this theorem alone, not requiring ax-8 1683 or excessive distinct variable conditions. (Contributed by Wolf Lammen, 12-Nov-2017.) (Proof shortened by Wolf Lammen, 9-Dec-2017.)
|- ph   =>    |- E. x ph
Theorem	exiftruOLD 1666	Obsolete proof of exiftru 1665 as of 9-Dec-2017. (Contributed by Wolf Lammen, 12-Nov-2017.) (New usage is discouraged.) (Proof modification is discouraged.)
|- ph   =>    |- E. x ph
Theorem	19.2 1667	Theorem 19.2 of [Margaris] p. 89. Note: This proof is very different from Margaris' because we only have Tarski's FOL axiom schemes available at this point. See the later 19.2g 1769 for a more conventional proof. Revised to remove dependency on ax-8 1683. (Contributed by NM, 2-Aug-2017.) (Revised by Wolf Lammen, 4-Dec-2017.)
|- ( A. x ph -> E. x ph )
Theorem	19.8w 1668	Weak version of 19.8a 1758. Uses only Tarski's FOL axiom schemes. (Contributed by NM, 1-Aug-2017.) (Proof shortened by Wolf Lammen, 4-Dec-2017.)
|- ( ph -> A. x ph )   =>    |- ( ph -> E. x ph )
Theorem	19.39 1669	Theorem 19.39 of [Margaris] p. 90. (Contributed by NM, 5-Aug-1993.)
|- ( ( E. x ph -> E. x ps ) -> E. x ( ph -> ps ) )
Theorem	19.24 1670	Theorem 19.24 of [Margaris] p. 90. (Contributed by NM, 5-Aug-1993.)
|- ( ( A. x ph -> A. x ps ) -> E. x ( ph -> ps ) )
Theorem	19.34 1671	Theorem 19.34 of [Margaris] p. 90. (Contributed by NM, 5-Aug-1993.)
|- ( ( A. x ph \/ E. x ps ) -> E. x ( ph \/ ps ) )
Theorem	19.9v 1672*	Special case of Theorem 19.9 of [Margaris] p. 89. Revised to remove dependency on ax-8 1683. (Contributed by NM, 28-May-1995.) (Revised by NM, 1-Aug-2017.) (Revised by Wolf Lammen, 4-Dec-2017.)
|- ( E. x ph <-> ph )
Theorem	19.3v 1673*	Special case of Theorem 19.3 of [Margaris] p. 89. Revised to remove dependency on ax-8 1683. (Contributed by NM, 1-Aug-2017.) (Revised by Wolf Lammen, 4-Dec-2017.)
|- ( A. x ph <-> ph )
Theorem	spvw 1674*	Version of sp 1759 when x does not occur in ph. This provides the other direction of ax-17 1623. Uses only Tarski's FOL axiom schemes. (Contributed by NM, 10-Apr-2017.) (Proof shortened by Wolf Lammen, 4-Dec-2017.)
|- ( A. x ph -> ph )
Theorem	spimeh 1675*	Existential introduction, using implicit substitution. Compare Lemma 14 of [Tarski] p. 70. (Contributed by NM, 7-Aug-1994.) (Proof shortened by Wolf Lammen, 10-Dec-2017.)
|- ( ph -> A. x ph )   &    |- ( x = z -> ( ph -> ps ) )   =>    |- ( ph -> E. x ps )
Theorem	spimw 1676*	Specialization. Lemma 8 of [KalishMontague] p. 87. Uses only Tarski's FOL axiom schemes. (Contributed by NM, 19-Apr-2017.) (Proof shortened by Wolf Lammen, 7-Aug-2017.)
|- ( -. ps -> A. x -. ps )   &    |- ( x = y -> ( ph -> ps ) )   =>    |- ( A. x ph -> ps )
Theorem	spimvw 1677*	Specialization. Lemma 8 of [KalishMontague] p. 87. Uses only Tarski's FOL axiom schemes. (Contributed by NM, 9-Apr-2017.)
|- ( x = y -> ( ph -> ps ) )   =>    |- ( A. x ph -> ps )
Theorem	spnfw 1678	Weak version of sp 1759. Uses only Tarski's FOL axiom schemes. (Contributed by NM, 1-Aug-2017.) (Proof shortened by Wolf Lammen, 13-Aug-2017.)
|- ( -. ph -> A. x -. ph )   =>    |- ( A. x ph -> ph )
Theorem	sptruw 1679	Version of sp 1759 when ph is true. Uses only Tarski's FOL axiom schemes. (Contributed by NM, 23-Apr-1017.)
|- ph   =>    |- ( A. x ph -> ph )
Theorem	spfalw 1680	Version of sp 1759 when ph is false. Uses only Tarski's FOL axiom schemes. (Contributed by NM, 23-Apr-1017.) (Proof shortened by Wolf Lammen, 25-Dec-2017.)
|- -. ph   =>    |- ( A. x ph -> ph )
Theorem	cbvaliw 1681*	Change bound variable. Uses only Tarski's FOL axiom schemes. Part of Lemma 7 of [KalishMontague] p. 86. (Contributed by NM, 19-Apr-2017.)
|- ( A. x ph -> A. y A. x ph )   &    |- ( -. ps -> A. x -. ps )   &    |- ( x = y -> ( ph -> ps ) )   =>    |- ( A. x ph -> A. y ps )
Theorem	cbvalivw 1682*	Change bound variable. Uses only Tarski's FOL axiom schemes. Part of Lemma 7 of [KalishMontague] p. 86. (Contributed by NM, 9-Apr-2017.)
|- ( x = y -> ( ph -> ps ) )   =>    |- ( A. x ph -> A. y ps )
1.4.7  Axiom scheme ax-8 (Equality)
Axiom	ax-8 1683	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 1690). 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 C7 of [Monk2] p. 105 and Axiom Scheme C8' in [Megill] p. 448 (p. 16 of the preprint). The equality symbol was invented in 1527 by Robert Recorde. He chose a pair of parallel lines of the same length because "noe .2. thynges, can be moare equalle." Note that this axiom is still valid even when any two or all three of x, y, and z are replaced with the same variable since they do not have any distinct variable (Metamath's $d) restrictions. Because of this, we say that these three variables are "bundled" (a term coined by Raph Levien). (Contributed by NM, 5-Aug-1993.)
|- ( x = y -> ( x = z -> y = z ) )
Theorem	equid 1684	Identity law for equality. Lemma 2 of [KalishMontague] p. 85. See also Lemma 6 of [Tarski] p. 68. (Contributed by NM, 1-Apr-2005.) (Revised by NM, 9-Apr-2017.) (Proof shortened by Wolf Lammen, 5-Feb-2018.)
|- x = x
Theorem	equidOLD 1685	Obsolete proof of equid 1684 as of 9-Dec-2017. (Contributed by NM, 1-Apr-2005.) (New usage is discouraged.) (Proof modification is discouraged.)
|- x = x
Theorem	nfequid 1686	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. (Contributed by NM, 13-Jan-2011.) (Revised by NM, 21-Aug-2017.)
|- F/ y x = x
Theorem	equcomi 1687	Commutative law for equality. Lemma 3 of [KalishMontague] p. 85. See also Lemma 7 of [Tarski] p. 69. (Contributed by NM, 5-Aug-1993.) (Revised by NM, 9-Apr-2017.)
|- ( x = y -> y = x )
Theorem	equcom 1688	Commutative law for equality. (Contributed by NM, 20-Aug-1993.)
|- ( x = y <-> y = x )
Theorem	equcoms 1689	An inference commuting equality in antecedent. Used to eliminate the need for a syllogism. (Contributed by NM, 5-Aug-1993.)
|- ( x = y -> ph )   =>    |- ( y = x -> ph )
Theorem	equtr 1690	A transitive law for equality. (Contributed by NM, 23-Aug-1993.)
|- ( x = y -> ( y = z -> x = z ) )
Theorem	equtrr 1691	A transitive law for equality. Lemma L17 in [Megill] p. 446 (p. 14 of the preprint). (Contributed by NM, 23-Aug-1993.)
|- ( x = y -> ( z = x -> z = y ) )
Theorem	equequ1 1692	An equivalence law for equality. (Contributed by NM, 5-Aug-1993.) (Proof shortened by Wolf Lammen, 10-Dec-2017.)
|- ( x = y -> ( x = z <-> y = z ) )
Theorem	equequ1OLD 1693	Obsolete version of equequ1 1692 as of 12-Nov-2017. (Contributed by NM, 5-Aug-1993.) (New usage is discouraged.) (Proof modification is discouraged.)
|- ( x = y -> ( x = z <-> y = z ) )
Theorem	equequ2 1694	An equivalence law for equality. (Contributed by NM, 5-Aug-1993.) (Proof shortened by Wolf Lammen, 4-Aug-2017.)
|- ( x = y -> ( z = x <-> z = y ) )
Theorem	stdpc6 1695	One of the two equality axioms of standard predicate calculus, called reflexivity of equality. (The other one is stdpc7 1938.) Axiom 6 of [Mendelson] p. 95. Mendelson doesn't say why he prepended the redundant quantifier, but it was probably to be compatible with free logic (which is valid in the empty domain). (Contributed by NM, 16-Feb-2005.)
|- A. x x = x
Theorem	equtr2 1696	A transitive law for equality. (Contributed by NM, 12-Aug-1993.) (Proof shortened by Andrew Salmon, 25-May-2011.)
|- ( ( x = z /\ y = z ) -> x = y )
Theorem	ax12b 1697	Two equivalent ways of expressing ax-12 1946. See the comment for ax-12 1946. (Contributed by NM, 2-May-2017.) (Proof shortened by Wolf Lammen, 26-Feb-2018.)
|- ( ( -. x = y -> ( y = z -> A. x y = z ) ) <-> ( -. x = y -> ( -. x = z -> ( y = z -> A. x y = z ) ) ) )
Theorem	ax12bOLD 1698	Obsolete version of ax12b 1697 as of 12-Aug-2017. (Contributed by NM, 2-May-2017.) (New usage is discouraged.) (Proof modification is discouraged.)
|- ( ( -. x = y -> ( y = z -> A. x y = z ) ) <-> ( -. x = y -> ( -. x = z -> ( y = z -> A. x y = z ) ) ) )
Theorem	spfw 1699*	Weak version of sp 1759. Uses only Tarski's FOL axiom schemes. Lemma 9 of [KalishMontague] p. 87. This may be the best we can do with minimal distinct variable conditions. TO DO: Do we need this theorem? If not, maybe it should be deleted. (Contributed by NM, 19-Apr-2017.)
|- ( -. ps -> A. x -. ps )   &    |- ( A. x ph -> A. y A. x ph )   &    |- ( -. ph -> A. y -. ph )   &    |- ( x = y -> ( ph <-> ps ) )   =>    |- ( A. x ph -> ph )
Theorem	spnfwOLD 1700	Weak version of sp 1759. Uses only Tarski's FOL axiom schemes. Obsolete version of spnfw 1678 as of 13-Aug-2017. (Contributed by NM, 1-Aug-2017.) (New usage is discouraged.) (Proof modification is discouraged.)
|- ( -. ph -> A. x -. ph )   =>    |- ( A. x ph -> ph )