mmtheorems16 - Metamath Proof Explorer

Jump to page: Contents + 1 1-100 2 101-200 3 201-300 4 301-400 5 401-500 6 501-600 7 601-700 8 701-800 9 801-900 10 901-1000 11 1001-1100 12 1101-1200 13 1201-1300 14 1301-1400 15 1401-1500 16 1501-1600 17 1601-1700 18 1701-1800 19 1801-1900 20 1901-2000 21 2001-2100 22 2101-2200 23 2201-2300 24 2301-2400 25 2401-2500 26 2501-2600 27 2601-2700 28 2701-2800 29 2801-2900 30 2901-3000 31 3001-3100 32 3101-3200 33 3201-3300 34 3301-3400 35 3401-3500 36 3501-3600 37 3601-3700 38 3701-3800 39 3801-3900 40 3901-4000 41 4001-4100 42 4101-4200 43 4201-4300 44 4301-4400 45 4401-4500 46 4501-4600 47 4601-4700 48 4701-4800 49 4801-4900 50 4901-5000 51 5001-5100 52 5101-5200 53 5201-5300 54 5301-5400 55 5401-5500 56 5501-5600 57 5601-5700 58 5701-5800 59 5801-5900 60 5901-6000 61 6001-6100 62 6101-6200 63 6201-6300 64 6301-6400 65 6401-6500 66 6501-6600 67 6601-6700 68 6701-6800 69 6801-6900 70 6901-7000 71 7001-7100 72 7101-7200 73 7201-7300 74 7301-7400 75 7401-7500 76 7501-7600 77 7601-7700 78 7701-7800 79 7801-7900 80 7901-8000 81 8001-8100 82 8101-8200 83 8201-8300 84 8301-8400 85 8401-8500 86 8501-8600 87 8601-8700 88 8701-8800 89 8801-8900 90 8901-9000 91 9001-9100 92 9101-9200 93 9201-9300 94 9301-9400 95 9401-9500 96 9501-9600 97 9601-9700 98 9701-9800 99 9801-9900 100 9901-10000 101 10001-10100 102 10101-10200 103 10201-10300 104 10301-10400 105 10401-10500 106 10501-10600 107 10601-10700 108 10701-10800 109 10801-10900 110 10901-11000 111 11001-11100 112 11101-11200 113 11201-11300 114 11301-11400 115 11401-11500 116 11501-11600 117 11601-11700 118 11701-11800 119 11801-11900 120 11901-12000 121 12001-12100 122 12101-12200 123 12201-12300 124 12301-12400 125 12401-12500 126 12501-12600 127 12601-12700 128 12701-12800 129 12801-12900 130 12901-13000 131 13001-13100 132 13101-13200 133 13201-13300 134 13301-13400 135 13401-13500 136 13501-13600 137 13601-13700 138 13701-13800 139 13801-13900 140 13901-14000 141 14001-14100 142 14101-14200 143 14201-14300 144 14301-14400 145 14401-14500 146 14501-14600 147 14601-14700 148 14701-14800 149 14801-14900 150 14901-15000 151 15001-15100 152 15101-15200 153 15201-15300 154 15301-15400 155 15401-15500 156 15501-15600 157 15601-15700 158 15701-15800 159 15801-15900 160 15901-16000 161 16001-16100 162 16101-16200 163 16201-16300 164 16301-16400 165 16401-16500 166 16501-16600 167 16601-16700 168 16701-16800 169 16801-16900 170 16901-17000 171 17001-17100 172 17101-17200 173 17201-17300 174 17301-17400 175 17401-17411

Statement List for Metamath Proof Explorer - 1501-1600 - Page 16 of 175

Type	Label	Description
Statement
Theorem	ax10 1501	Rederivation of ax-10 1308 from original version ax-10o 1500. See theorem ax10o 1499 for the derivation of ax-10o 1500 from ax-10 1308. This theorem should not be referenced in any proof. Instead, use ax-10 1308 above so that uses of ax-10 1308 can be more easily identified.
|- (A.x x = y -> A.y y = x)
Theorem	alequcom 1502	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).
|- (A.x x = y -> A.y y = x)
Theorem	alequcoms 1503	A commutation rule for identical variable specifiers.
|- (A.x x = y -> ph)   =>   |- (A.y y = x -> ph)
Theorem	nalequcoms 1504	A commutation rule for distinct variable specifiers.
|- (-. A.x x = y -> ph)   =>   |- (-. A.y y = x -> ph)
Theorem	hbae 1505	All variables are effectively bound in an identical variable specifier.
|- (A.x x = y -> A.zA.x x = y)
Theorem	hbaes 1506	Rule that applies hbae 1505 to antecedent.
|- (A.zA.x x = y -> ph)   =>   |- (A.x x = y -> ph)
Theorem	hbnae 1507	All variables are effectively bound in a distinct variable specifier. Lemma L19 in [Megill] p. 446 (p. 14 of the preprint).
|- (-. A.x x = y -> A.z -. A.x x = y)
Theorem	hbnaes 1508	Rule that applies hbnae 1507 to antecedent.
|- (A.z -. A.x x = y -> ph)   =>   |- (-. A.x x = y -> ph)
Theorem	equs3 1509	Lemma used in proofs of substitution properties.
|- (E.x(x = y /\ ph) <-> -. A.x(x = y -> -. ph))
Theorem	equs4 1510	Lemma used in proofs of substitution properties.
|- (A.x(x = y -> ph) -> E.x(x = y /\ ph))
Theorem	equsal 1511	A useful equivalence related to substitution. (The proof was shortened by Andrew Salmon, 12-Aug-2011.)
|- (ps -> A.xps)   &   |- (x = y -> (ph <-> ps))   =>   |- (A.x(x = y -> ph) <-> ps)
Theorem	equsalOLD 1512	A useful equivalence related to substitution.
|- (ps -> A.xps)   &   |- (x = y -> (ph <-> ps))   =>   |- (A.x(x = y -> ph) <-> ps)
Theorem	equsex 1513	A useful equivalence related to substitution.
|- (ps -> A.xps)   &   |- (x = y -> (ph <-> ps))   =>   |- (E.x(x = y /\ ph) <-> ps)
Theorem	dvelimfALT 1514	Proof of dvelimf 1623 without using ax-11 1309. See dvelimALT 1744 for a proof (of the distinct variable version dvelim 1743) that doesn't require ax-10 1308.
|- (ph -> A.xph)   &   |- (ps -> A.zps)   &   |- (z = y -> (ph <-> ps))   =>   |- (-. A.x x = y -> (ps -> A.xps))
Theorem	dral1 1515	Formula-building lemma for use with the Distinctor Reduction Theorem. Part of Theorem 9.4 of [Megill] p. 448 (p. 16 of preprint).
|- (A.x x = y -> (ph <-> ps))   =>   |- (A.x x = y -> (A.xph <-> A.yps))
Theorem	dral2 1516	Formula-building lemma for use with the Distinctor Reduction Theorem. Part of Theorem 9.4 of [Megill] p. 448 (p. 16 of preprint).
|- (A.x x = y -> (ph <-> ps))   =>   |- (A.x x = y -> (A.zph <-> A.zps))
Theorem	drex1 1517	Formula-building lemma for use with the Distinctor Reduction Theorem. Part of Theorem 9.4 of [Megill] p. 448 (p. 16 of preprint).
|- (A.x x = y -> (ph <-> ps))   =>   |- (A.x x = y -> (E.xph <-> E.yps))
Theorem	drex2 1518	Formula-building lemma for use with the Distinctor Reduction Theorem. Part of Theorem 9.4 of [Megill] p. 448 (p. 16 of preprint).
|- (A.x x = y -> (ph <-> ps))   =>   |- (A.x x = y -> (E.zph <-> E.zps))
Theorem	a4imt 1519	Closed theorem form of a4im 1520.
|- (A.x((ps -> A.xps) /\ (x = y -> (ph -> ps))) -> (A.xph -> ps))
Theorem	a4im 1520	Specialization, using implicit substitition. Compare Lemma 14 of [Tarski] p. 70. The a4im 1520 series of theorems requires that only one direction of the substitution hypothesis hold.
|- (ps -> A.xps)   &   |- (x = y -> (ph -> ps))   =>   |- (A.xph -> ps)
Theorem	a4ime 1521	Existential introduction, using implicit substitition. Compare Lemma 14 of [Tarski] p. 70.
|- (ph -> A.xph)   &   |- (x = y -> (ph -> ps))   =>   |- (ph -> E.xps)
Theorem	a4imed 1522	Deduction version of a4ime 1521.
|- (ch -> A.xch)   &   |- (ch -> (ph -> A.xph))   &   |- (x = y -> (ph -> ps))   =>   |- (ch -> (ph -> E.xps))
Theorem	cbv1 1523	Rule used to change bound variables, using implicit substitition.
|- (ph -> (ps -> A.yps))   &   |- (ph -> (ch -> A.xch))   &   |- (ph -> (x = y -> (ps -> ch)))   =>   |- (A.xA.yph -> (A.xps -> A.ych))
Theorem	cbv2 1524	Rule used to change bound variables, using implicit substitition.
|- (ph -> (ps -> A.yps))   &   |- (ph -> (ch -> A.xch))   &   |- (ph -> (x = y -> (ps <-> ch)))   =>   |- (A.xA.yph -> (A.xps <-> A.ych))
Theorem	cbv3 1525	Rule used to change bound variables, using implicit substitition, that does not use ax-12 1310.
|- (ph -> A.yph)   &   |- (ps -> A.xps)   &   |- (x = y -> (ph -> ps))   =>   |- (A.xph -> A.yps)
Theorem	cbv3ALT 1526	Rule used to change bound variables, using implicit substitition. (The proof was shortened by Andrew Salmon, 25-May-2011.)
|- (ph -> A.yph)   &   |- (ps -> A.xps)   &   |- (x = y -> (ph -> ps))   =>   |- (A.xph -> A.yps)
Theorem	cbval 1527	Rule used to change bound variables, using implicit substitition. (The proof was shortened by Andrew Salmon, 25-May-2011.)
|- (ph -> A.yph)   &   |- (ps -> A.xps)   &   |- (x = y -> (ph <-> ps))   =>   |- (A.xph <-> A.yps)
Theorem	cbvalOLD 1528	Rule used to change bound variables, using implicit substitition.
|- (ph -> A.yph)   &   |- (ps -> A.xps)   &   |- (x = y -> (ph <-> ps))   =>   |- (A.xph <-> A.yps)
Theorem	cbvex 1529	Rule used to change bound variables, using implicit substitition.
|- (ph -> A.yph)   &   |- (ps -> A.xps)   &   |- (x = y -> (ph <-> ps))   =>   |- (E.xph <-> E.yps)
Theorem	chvar 1530	Implicit substitution of y for x into a theorem. (Contributed by Raph Levien, 9-Jul-2003.)
|- (ps -> A.xps)   &   |- (x = y -> (ph <-> ps))   &   |- ph   =>   |- ps
Theorem	equvini 1531	A variable introduction law for equality. Lemma 15 of [Monk2] p. 109, however we do not require z to be distinct from x and y (making the proof longer). (The proof was shortened by Andrew Salmon, 25-May-2011.)
|- (x = y -> E.z(x = z /\ z = y))
Theorem	equviniOLD 1532	A variable introduction law for equality. Lemma 15 of [Monk2] p. 109, however we do not require z to be distinct from x and y (making the proof longer).
|- (x = y -> E.z(x = z /\ z = y))
Theorem	hbequid2 1533	Bound-variable hypothesis builder for x = x. This theorem tells us that x is effectively not free in x = x, even though it is technically free according to the traditional definition of free variable. (The proof shows that this can be proved without ax-9 1307, even though the theorem equid 1484 cannot be. A shorter proof that uses ax-9 1307 is obtainable from equid 1484 and hbth 1348.) See hbequid 1313 for a more general version.
|- (x = x -> A.x x = x)
Substitution (without distinct variables)
Syntax	wsbc 1534	Extend wff notation to include the proper substitution of a class for a set. Read this notation as "the proper substitution of class A for set variable x in wff ph." (The purpose of introducing wff [A / x]ph here is to allow us to express i.e. "prove" the wsb 1535 of predicate calculus in terms of the wsbc 1534 of set theory, so that we don't "overload" its connectives with two syntax definitions. This is done to prevent ambiguity that would complicate some Metamath parsers. The class variable A is introduced temporarily for the purpose of this definition but otherwise not used in predicate calculus. See df-sbc 2454 for more information on the set theory usage of wsbc 1534.)
wff [A / x]ph
Theorem	wsb 1535	Extend wff definition to include proper substitution (read "the wff that results when y is properly substituted for x in wff ph"). (Instead of introducing wsb 1535 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 wsbc 1534. This lets us avoid overloading its connectives, thus preventing ambiguity that would complicate some Metamath parsers. Note: To see the proof steps of this syntax proof, type "show proof wsb /all" in the Metamath program.)
wff [y / x]ph
Definition	df-sb 1536	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 when y is properly substituted 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 1550. 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 remarkable little formula that is exactly equivalent and gives us a single direct definition. We later prove that our definition has the properties we expect of proper substitution (see theorems sbequ 1599, sbcom2 1724 and sbid2v 1734). Note that our definition is valid even when x and y are replaced with the same variable, as sbid 1549 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 1730 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 1596. When x and y are distinct, we can express proper substitution with the simpler expressions of sb5 1645 and sb6 1644. There are no restrictions on any of the variables, including what variables may occur in wff ph.
|- ([y / x]ph <-> ((x = y -> ph) /\ E.x(x = y /\ ph)))
Theorem	sbimi 1537	Infer substitution into antecedent and consequent of an implication.
|- (ph -> ps)   =>   |- ([y / x]ph -> [y / x]ps)
Theorem	sbbii 1538	Infer substitution into both sides of a logical equivalence.
|- (ph <-> ps)   =>   |- ([y / x]ph <-> [y / x]ps)
Theorem	drsb1 1539	Formula-building lemma for use with the Distinctor Reduction Theorem. Part of Theorem 9.4 of [Megill] p. 448 (p. 16 of preprint).
|- (A.x x = y -> ([z / x]ph <-> [z / y]ph))
Theorem	sb1 1540	One direction of a simplified definition of substitution.
|- ([y / x]ph -> E.x(x = y /\ ph))
Theorem	sb2 1541	One direction of a simplified definition of substitution.
|- (A.x(x = y -> ph) -> [y / x]ph)
Theorem	sbequ1 1542	An equality theorem for substitution.
|- (x = y -> (ph -> [y / x]ph))
Theorem	sbequ2 1543	An equality theorem for substitution.
|- (x = y -> ([y / x]ph -> ph))
Theorem	stdpc7 1544	One of the two equality axioms of standard predicate calculus, called substitutivity of equality. (The other one is stdpc6 1486.) Translated to traditional notation, it can be read: "x = y -> (ph(x, x) -> ph(x, y)), provided that y is free for x in ph(x, y)." Axiom 7 of [Mendelson] p. 95.
|- (x = y -> ([x / y]ph -> ph))
Theorem	sbequ12 1545	An equality theorem for substitution.
|- (x = y -> (ph <-> [y / x]ph))
Theorem	sbequ12r 1546	An equality theorem for substitution. (The proof was shortened by Andrew Salmon, 21-Jun-2011.)
|- (x = y -> ([x / y]ph <-> ph))
Theorem	sbequ12rOLD 1547	An equality theorem for substitution.
|- (x = y -> ([x / y]ph <-> ph))
Theorem	sbequ12a 1548	An equality theorem for substitution.
|- (x = y -> ([y / x]ph <-> [x / y]ph))
Theorem	sbid 1549	An identity theorem for substitution. Remark 9.1 in [Megill] p. 447 (p. 15 of the preprint).
|- ([x / x]ph <-> ph)
Theorem	stdpc4 1550	The specialization axiom of standard predicate calculus. It states that if a statement ph holds for all x, then it also holds for the specific case of y (properly) substituted for x. Translated to traditional notation, it can be read: "A.xph(x) -> ph(y), provided that y is free for x in ph(x)." Axiom 4 of [Mendelson] p. 69. See also a4sbc 2457 and ra4sbc 2536.
|- (A.xph -> [y / x]ph)
Theorem	sbf 1551	Substitution for a variable not free in a wff does not affect it.
|- (ph -> A.xph)   =>   |- ([y / x]ph <-> ph)
Theorem	sbf2 1552	Substitution has no effect on a bound variable.
|- ([y / x]A.xph <-> A.xph)
Theorem	sb6x 1553	Equivalence involving substitution for a variable not free. (The proof was shortened by Andrew Salmon, 12-Aug-2011.)
|- (ph -> A.xph)   =>   |- ([y / x]ph <-> A.x(x = y -> ph))
Theorem	sb6xOLD 1554	Equivalence involving substitution for a variable not free.
|- (ph -> A.xph)   =>   |- ([y / x]ph <-> A.x(x = y -> ph))
Theorem	hbs1f 1555	If x is not free in ph, it is not free in [y / x]ph. (The proof was shortened by Andrew Salmon, 25-May-2011.)
|- (ph -> A.xph)   =>   |- ([y / x]ph -> A.x[y / x]ph)
Theorem	hbs1fOLD 1556	If x is not free in ph, it is not free in [y / x]ph.
|- (ph -> A.xph)   =>   |- ([y / x]ph -> A.x[y / x]ph)
Theorem	sbequ5 1557	Substitution does not change an identical variable specifier.
|- ([w / z]A.x x = y <-> A.x x = y)
Theorem	sbequ6 1558	Substitution does not change a distinctor.
|- ([w / z] -. A.x x = y <-> -. A.x x = y)
Theorem	sbt 1559	A substitution into a theorem remains true. (See chvar 1530 and chvarv 1712 for versions, using implicit substitition.) (The proof was shortened by Andrew Salmon, 25-May-2011.)
|- ph   =>   |- [y / x]ph
Theorem	sbtOLD 1560	A substitution into a theorem remains true. (See chvar 1530 and chvarv 1712 for versions, using implicit substitition.)
|- ph   =>   |- [y / x]ph
Theorem	equsb1 1561	Substitution applied to an atomic wff.
|- [y / x]x = y
Theorem	equsb2 1562	Substitution applied to an atomic wff.
|- [y / x]y = x
Theorem	sbied 1563	Conversion of implicit substitution to explicit substitution (deduction version of sbie 1565). (The proof was shortened by Andrew Salmon, 25-May-2011.)
|- (ph -> A.xph)   &   |- (ph -> (ch -> A.xch))   &   |- (ph -> (x = y -> (ps <-> ch)))   =>   |- (ph -> ([y / x]ps <-> ch))
Theorem	sbiedOLD 1564	Conversion of implicit substitution to explicit substitution (deduction version of sbie 1565).
|- (ph -> A.xph)   &   |- (ph -> (ch -> A.xch))   &   |- (ph -> (x = y -> (ps <-> ch)))   =>   |- (ph -> ([y / x]ps <-> ch))
Theorem	sbie 1565	Conversion of implicit substitution to explicit substitution.
|- (ps -> A.xps)   &   |- (x = y -> (ph <-> ps))   =>   |- ([y / x]ph <-> ps)
Theorems using axiom ax-11
Theorem	equs5a 1566	A property related to substitution that unlike equs5 1591 doesn't require a distinctor antecedent.
|- (E.x(x = y /\ A.yph) -> A.x(x = y -> ph))
Theorem	equs5e 1567	A property related to substitution that unlike equs5 1591 doesn't require a distinctor antecedent.
|- (E.x(x = y /\ ph) -> A.x(x = y -> E.yph))
Theorem	sb4a 1568	A version of sb4 1593 that doesn't require a distinctor antecedent.
|- ([y / x]A.yph -> A.x(x = y -> ph))
Theorem	equs45f 1569	Two ways of expressing substitution when y is not free in ph.
|- (ph -> A.yph)   =>   |- (E.x(x = y /\ ph) <-> A.x(x = y -> ph))
Theorem	sb6f 1570	Equivalence for substitution when y is not free in ph.
|- (ph -> A.yph)   =>   |- ([y / x]ph <-> A.x(x = y -> ph))
Theorem	sb5f 1571	Equivalence for substitution when y is not free in ph.
|- (ph -> A.yph)   =>   |- ([y / x]ph <-> E.x(x = y /\ ph))
Theorem	sb4e 1572	One direction of a simplified definition of substitution that unlike sb4 1593 doesn't require a distinctor antecedent.
|- ([y / x]ph -> A.x(x = y -> E.yph))
Theorem	hbsb2a 1573	Special case of a bound-variable hypothesis builder for substitution.
|- ([y / x]A.yph -> A.x[y / x]ph)
Theorem	hbsb2e 1574	Special case of a bound-variable hypothesis builder for substitution.
|- ([y / x]ph -> A.x[y / x]E.yph)
Theorem	hbsb3 1575	If y is not free in ph, x is not free in [y / x]ph.
|- (ph -> A.yph)   =>   |- ([y / x]ph -> A.x[y / x]ph)
Predicate calculus with distinct variables
The axiom of quantifier introduction ax-17
Theorem	a4imv 1576	A version of a4im 1520 with a distinct variable requirement instead of a bound variable hypothesis.
|- (x = y -> (ph -> ps))   =>   |- (A.xph -> ps)
Theorem	aev 1577	A "distinctor elimination" lemma with no restrictions on variables in the consequent, proved without using ax-16 1580. (The proof was shortened by Andrew Salmon, 21-Jun-2011.)
|- (A.x x = y -> A.z w = v)
Theorem	aevOLD 1578	A "distinctor elimination" lemma with no restrictions on variables in the consequent, proved without using ax-16 1580. The proof is unusual in that it involves linking 17 implications, which might provide an interesting challenge for an automated theorem prover.
|- (A.x x = y -> A.z w = v)
Derive the axiom of distinct variables ax-16
Theorem	ax16 1579	Theorem showing that ax-16 1580 is redundant if ax-17 1317 is included in the axiom system. The important part of the proof is provided by aev 1577. See ax16ALT 1648 for an alternate proof that does not require ax-10 1308 or ax-12 1310. This theorem should not be referenced in any proof. Instead, use ax-16 1580 below so that theorems needing ax-16 1580 can be more easily identified.
|- (A.x x = y -> (ph -> A.xph))
Axiom	ax-16 1580	Axiom of Distinct Variables. The only axiom of predicate calculus requiring that variables be distinct (if we consider ax-17 1317 to be a metatheorem and not an axiom). Axiom scheme C16' 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. It is a somewhat bizarre axiom since the antecedent is always false in set theory (see dtru 3498), but nonetheless it is technically necessary as you can see from its uses. This axiom is redundant if we include ax-17 1317; see theorem ax16 1579. Alternately, ax-17 1317 becomes logically redundant in the presence of this axiom, but without ax-17 1317 we lose the more powerful metalogic that results from being able to express the concept of a set variable not occurring in a wff (as opposed to just two set variables being distinct). We retain ax-16 1580 here to provide logical completeness for systems with the simpler metalogic that results from omitting ax-17 1317, which might be easier to study for some theoretical purposes.
|- (A.x x = y -> (ph -> A.xph))
Theorem	ax17eq 1581	Theorem to add distinct quantifier to atomic formula. (This theorem demonstrates the induction basis for ax-17 1317 considered as a metatheorem. Do not use it for later proofs - use ax-17 1317 instead, to avoid reference to the redundant axiom ax-16 1580.)
|- (x = y -> A.z x = y)
Theorem	dveeq2 1582	Quantifier introduction when one pair of variables is distinct.
|- (-. A.x x = y -> (z = y -> A.x z = y))
Theorem	dveeq2ALT 1583	Version of dveeq2 1582 using ax-16 1580 instead of ax-17 1317.
|- (-. A.x x = y -> (z = y -> A.x z = y))
Theorem	19.23adv 1584	Deduction from Theorem 19.23 of [Margaris] p. 90.
|- (ph -> (ps -> ch))   =>   |- (ph -> (E.xps -> ch))
Theorem	ax11v2 1585	Recovery of ax11o 1587 from ax11v 1642 without using ax-11 1309. The hypothesis is even weaker than ax11v 1642, with z both distinct from x and not occurring in ph. Thus the hypothesis provides an alternate axiom that can be used in place of ax11o 1587.
|- (x = z -> (ph -> A.x(x = z -> ph)))   =>   |- (-. A.x x = y -> (x = y -> (ph -> A.x(x = y -> ph))))
Theorem	ax11a2 1586	Derive ax-11o 1588 from a hypothesis in the form of ax-11 1309. The hypothesis is even weaker than ax-11 1309, with z both distinct from x and not occurring in ph. Thus the hypothesis provides an alternate axiom that can be used in place of ax11o 1587. As theorem ax11 1589 shows, the distinct variable conditions are optional. An open problem is whether ax11o 1587 can be derived from ax-11 1309 without relying on ax-17 1317.
|- (x = z -> (A.zph -> A.x(x = z -> ph)))   =>   |- (-. A.x x = y -> (x = y -> (ph -> A.x(x = y -> ph))))
Derive the original axiom of variable substitution ax-11o
Theorem	ax11o 1587	Derivation of set.mm's original ax-11o 1588 from the shorter ax-11 1309 that has replaced it. An open problem is whether this theorem can be proved without relying on ax-16 1580 or ax-17 1317. Another open problem is whether this theorem can be proved without relying on ax-12 1310 (see note in a12study 1769). Theorem ax11 1589 shows the reverse derivation of ax-11 1309 from ax-11o 1588. This theorem should not be referenced in any proof. Instead, use ax-11o 1588 below so that theorems needing ax-11o 1588 can be more easily identified.
|- (-. A.x x = y -> (x = y -> (ph -> A.x(x = y -> ph))))
Axiom	ax-11o 1588	Axiom ax-11o 1588 ("o" for "old") was the original version of ax-11 1309, before it was discovered (in Jan. 2007) that the shorter ax-11 1309 could replace it. It appears as Axiom scheme C15' in [Megill] p. 448 (p. 16 of the preprint). 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. To understand this theorem more easily, think of "-. A.xx = y ->..." as informally meaning "if x and y are distinct variables then..." The antecedent becomes false if the same variable is substituted for x and y, ensuring the theorem is sound whenever this is the case. In some later theorems, we call an antecedent of the form -. A.xx = y a "distinctor." This axiom is redundant, as shown by theorem ax11o 1587.
|- (-. A.x x = y -> (x = y -> (ph -> A.x(x = y -> ph))))
Theorem	ax11 1589	Rederivation of axiom ax-11 1309 from the orginal version, ax-11o 1588. See theorem ax11o 1587 for the derivation of ax-11o 1588 from ax-11 1309. This theorem should not be referenced in any proof. Instead, use ax-11 1309 above so that uses of ax-11 1309 can be more easily identified.
|- (x = y -> (A.yph -> A.x(x = y -> ph)))
Theorems without distinct variables that use axiom ax-11o
Theorem	ax11b 1590	A bidirectional version of ax-11o 1588.
|- ((-. A.x x = y /\ x = y) -> (ph <-> A.x(x = y -> ph)))
Theorem	equs5 1591	Lemma used in proofs of substitution properties.
|- (-. A.x x = y -> (E.x(x = y /\ ph) -> A.x(x = y -> ph)))
Theorem	sb3 1592	One direction of a simplified definition of substitution when variables are distinct.
|- (-. A.x x = y -> (E.x(x = y /\ ph) -> [y / x]ph))
Theorem	sb4 1593	One direction of a simplified definition of substitution when variables are distinct.
|- (-. A.x x = y -> ([y / x]ph -> A.x(x = y -> ph)))
Theorem	sb4b 1594	Simplified definition of substitution when variables are distinct.
|- (-. A.x x = y -> ([y / x]ph <-> A.x(x = y -> ph)))
Theorem	dfsb2 1595	An alternate definition of proper substitution that, like df-sb 1536, mixes free and bound variables to avoid distinct variable requirements.
|- ([y / x]ph <-> ((x = y /\ ph) \/ A.x(x = y -> ph)))
Theorem	dfsb3 1596	An alternate definition of proper substitution df-sb 1536 that uses only primitive connectives (no defined terms) on the right-hand side.
|- ([y / x]ph <-> ((x = y -> -. ph) -> A.x(x = y -> ph)))
Theorem	hbsb2 1597	Bound-variable hypothesis builder for substitution.
|- (-. A.x x = y -> ([y / x]ph -> A.x[y / x]ph))
Theorem	sbequi 1598	An equality theorem for substitution.
|- (x = y -> ([x / z]ph -> [y / z]ph))
Theorem	sbequ 1599	An equality theorem for substitution. Used in proof of Theorem 9.7 in [Megill] p. 449 (p. 16 of the preprint).
|- (x = y -> ([x / z]ph <-> [y / z]ph))
Theorem	drsb2 1600	Formula-building lemma for use with the Distinctor Reduction Theorem. Part of Theorem 9.4 of [Megill] p. 448 (p. 16 of preprint).
|- (A.x x = y -> ([x / z]ph <-> [y / z]ph))