mmtheorems19 - Metamath Proof Explorer

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Statement List for Metamath Proof Explorer - 1801-1900 - Page 19 of 191

Type	Label	Description
Statement
Theorem	a4im 1801	Specialization, using implicit substitition. Compare Lemma 14 of [Tarski] p. 70. The a4im 1801 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 1802	Existential introduction, using implicit substitition. Compare Lemma 14 of [Tarski] p. 70.
|- (ph -> A.xph)   &   |- (x = y -> (ph -> ps))   =>   |- (ph -> E.xps)
Theorem	a4imed 1803	Deduction version of a4ime 1802.
|- (ch -> A.xch)   &   |- (ch -> (ph -> A.xph))   &   |- (x = y -> (ph -> ps))   =>   |- (ch -> (ph -> E.xps))
Theorem	cbv1 1804	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 1805	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 1806	Rule used to change bound variables, using implicit substitition, that does not use ax-12 1598.
|- (ph -> A.yph)   &   |- (ps -> A.xps)   &   |- (x = y -> (ph -> ps))   =>   |- (A.xph -> A.yps)
Theorem	cbv3ALT 1807	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 1808	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	cbvex 1809	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 1810	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 1811	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	equveli 1812	A variable elimination law for equality with no distinct variable requirements. (Compare equvini 1811.)
|- (A.z(z = x <-> z = y) -> x = y)
Theorem	hbequid2 1813	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 1595, even though the theorem equid 1766 cannot be. A shorter proof that uses ax-9 1595 is obtainable from equid 1766 and hbth 1637.) See hbequid 1601 for a more general version.
|- (x = x -> A.x x = x)
Substitution (without distinct variables)
Syntax	wsbc 1814	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 1815 of predicate calculus in terms of the wsbc 1814 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 2700 for more information on the set theory usage of wsbc 1814.)
wff [A / x]ph
Theorem	wsb 1815	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 1815 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 1814. 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 1816	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 1829. 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 1875, sbcom2 1990 and sbid2v 1998). Note that our definition is valid even when x and y are replaced with the same variable, as sbid 1828 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 1995 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 1872. When x and y are distinct, we can express proper substitution with the simpler expressions of sb5 1915 and sb6 1914. 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 1817	Infer substitution into antecedent and consequent of an implication.
|- (ph -> ps)   =>   |- ([y / x]ph -> [y / x]ps)
Theorem	sbbii 1818	Infer substitution into both sides of a logical equivalence.
|- (ph <-> ps)   =>   |- ([y / x]ph <-> [y / x]ps)
Theorem	drsb1 1819	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 1820	One direction of a simplified definition of substitution.
|- ([y / x]ph -> E.x(x = y /\ ph))
Theorem	sb2 1821	One direction of a simplified definition of substitution.
|- (A.x(x = y -> ph) -> [y / x]ph)
Theorem	sbequ1 1822	An equality theorem for substitution.
|- (x = y -> (ph -> [y / x]ph))
Theorem	sbequ2 1823	An equality theorem for substitution.
|- (x = y -> ([y / x]ph -> ph))
Theorem	stdpc7 1824	One of the two equality axioms of standard predicate calculus, called substitutivity of equality. (The other one is stdpc6 1768.) 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 1825	An equality theorem for substitution.
|- (x = y -> (ph <-> [y / x]ph))
Theorem	sbequ12r 1826	An equality theorem for substitution. (The proof was shortened by Andrew Salmon, 21-Jun-2011.)
|- (x = y -> ([x / y]ph <-> ph))
Theorem	sbequ12a 1827	An equality theorem for substitution.
|- (x = y -> ([y / x]ph <-> [x / y]ph))
Theorem	sbid 1828	An identity theorem for substitution. Remark 9.1 in [Megill] p. 447 (p. 15 of the preprint).
|- ([x / x]ph <-> ph)
Theorem	stdpc4 1829	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 2703 and ra4sbc 2767.
|- (A.xph -> [y / x]ph)
Theorem	sbf 1830	Substitution for a variable not free in a wff does not affect it.
|- (ph -> A.xph)   =>   |- ([y / x]ph <-> ph)
Theorem	sbf2 1831	Substitution has no effect on a bound variable.
|- ([y / x]A.xph <-> A.xph)
Theorem	sb6x 1832	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	hbs1f 1833	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	sbequ5 1834	Substitution does not change an identical variable specifier.
|- ([w / z]A.x x = y <-> A.x x = y)
Theorem	sbequ6 1835	Substitution does not change a distinctor.
|- ([w / z] -. A.x x = y <-> -. A.x x = y)
Theorem	sbt 1836	A substitution into a theorem remains true. (See chvar 1810 and chvarv 1978 for versions using implicit substitition.) (The proof was shortened by Andrew Salmon, 25-May-2011.)
|- ph   =>   |- [y / x]ph
Theorem	equsb1 1837	Substitution applied to an atomic wff.
|- [y / x]x = y
Theorem	equsb2 1838	Substitution applied to an atomic wff.
|- [y / x]y = x
Theorem	sbied 1839	Conversion of implicit substitution to explicit substitution (deduction version of sbie 1840). (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	sbie 1840	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 1841	A property related to substitution that unlike equs5 1867 doesn't require a distinctor antecedent.
|- (E.x(x = y /\ A.yph) -> A.x(x = y -> ph))
Theorem	equs5e 1842	A property related to substitution that unlike equs5 1867 doesn't require a distinctor antecedent.
|- (E.x(x = y /\ ph) -> A.x(x = y -> E.yph))
Theorem	sb4a 1843	A version of sb4 1869 that doesn't require a distinctor antecedent.
|- ([y / x]A.yph -> A.x(x = y -> ph))
Theorem	equs45f 1844	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 1845	Equivalence for substitution when y is not free in ph.
|- (ph -> A.yph)   =>   |- ([y / x]ph <-> A.x(x = y -> ph))
Theorem	sb5f 1846	Equivalence for substitution when y is not free in ph.
|- (ph -> A.yph)   =>   |- ([y / x]ph <-> E.x(x = y /\ ph))
Theorem	sb4e 1847	One direction of a simplified definition of substitution that unlike sb4 1869 doesn't require a distinctor antecedent.
|- ([y / x]ph -> A.x(x = y -> E.yph))
Theorem	hbsb2a 1848	Special case of a bound-variable hypothesis builder for substitution.
|- ([y / x]A.yph -> A.x[y / x]ph)
Theorem	hbsb2e 1849	Special case of a bound-variable hypothesis builder for substitution.
|- ([y / x]ph -> A.x[y / x]E.yph)
Theorem	hbsb3 1850	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
Uses of the axiom of quantifier introduction ax-17
Theorem	a4imv 1851	A version of a4im 1801 with a distinct variable requirement instead of a bound variable hypothesis.
|- (x = y -> (ph -> ps))   =>   |- (A.xph -> ps)
Theorem	aev 1852	A "distinctor elimination" lemma with no restrictions on variables in the consequent, proved without using ax-16 1854. (The proof was shortened by Andrew Salmon, 21-Jun-2011.)
|- (A.x x = y -> A.z w = v)
Derive the axiom of distinct variables ax-16
Theorem	ax16 1853	Theorem showing that ax-16 1854 is redundant if ax-17 1605 is included in the axiom system. The important part of the proof is provided by aev 1852. See ax16ALT 1918 for an alternate proof that does not require ax-10 1596 or ax-12 1598. This theorem should not be referenced in any proof. Instead, use ax-16 1854 below so that theorems needing ax-16 1854 can be more easily identified.
|- (A.x x = y -> (ph -> A.xph))
Axiom	ax-16 1854	Axiom of Distinct Variables. The only axiom of predicate calculus requiring that variables be distinct (if we consider ax-17 1605 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 3662), but nonetheless it is technically necessary as you can see from its uses. This axiom is redundant if we include ax-17 1605; see theorem ax16 1853. Alternately, ax-17 1605 becomes logically redundant in the presence of this axiom, but without ax-17 1605 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 1854 here to provide logical completeness for systems with the simpler metalogic that results from omitting ax-17 1605, which might be easier to study for some theoretical purposes.
|- (A.x x = y -> (ph -> A.xph))
Theorem	ax17eq 1855	Theorem to add distinct quantifier to atomic formula. (This theorem demonstrates the induction basis for ax-17 1605 considered as a metatheorem. Do not use it for later proofs - use ax-17 1605 instead, to avoid reference to the redundant axiom ax-16 1854.)
|- (x = y -> A.z x = y)
Theorem	dveeq2 1856	Quantifier introduction when one pair of variables is distinct.
|- (-. A.x x = y -> (z = y -> A.x z = y))
Theorem	dveeq2ALT 1857	Version of dveeq2 1856 using ax-16 1854 instead of ax-17 1605.
|- (-. A.x x = y -> (z = y -> A.x z = y))
Theorem	dvelimfALT2 1858	Proof of dvelimf 1897 using dveeq2 1856 instead of ax-12 1598. This shows that ax-12 1598 could be replaced by dveeq2 1856. (Contributed by Andrew Salmon, 21-Jul-2011.)
|- (ph -> A.xph)   &   |- (ps -> A.zps)   &   |- (z = y -> (ph <-> ps))   =>   |- (-. A.x x = y -> (ps -> A.xps))
Theorem	nd5 1859	A lemma for proving conditionless ZFC axioms.
|- (-. A.y y = x -> (z = y -> A.x z = y))
Theorem	19.23adv 1860	Deduction from Theorem 19.23 of [Margaris] p. 90.
|- (ph -> (ps -> ch))   =>   |- (ph -> (E.xps -> ch))
Theorem	ax11v2 1861	Recovery of ax11o 1863 from ax11v 1912 without using ax-11 1597. The hypothesis is even weaker than ax11v 1912, 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 1863.
|- (x = z -> (ph -> A.x(x = z -> ph)))   =>   |- (-. A.x x = y -> (x = y -> (ph -> A.x(x = y -> ph))))
Theorem	ax11a2 1862	Derive ax-11o 1864 from a hypothesis in the form of ax-11 1597. The hypothesis is even weaker than ax-11 1597, 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 1863. As theorem ax11 1865 shows, the distinct variable conditions are optional. An open problem is whether ax11o 1863 can be derived from ax-11 1597 without relying on ax-17 1605.
|- (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 1863	Derivation of set.mm's original ax-11o 1864 from the shorter ax-11 1597 that has replaced it. An open problem is whether this theorem can be proved without relying on ax-16 1854 or ax-17 1605. Another open problem is whether this theorem can be proved without relying on ax-12 1598 (see note in a12study 2033). Theorem ax11 1865 shows the reverse derivation of ax-11 1597 from ax-11o 1864. This theorem should not be referenced in any proof. Instead, use ax-11o 1864 below so that theorems needing ax-11o 1864 can be more easily identified.
|- (-. A.x x = y -> (x = y -> (ph -> A.x(x = y -> ph))))
Axiom	ax-11o 1864	Axiom ax-11o 1864 ("o" for "old") was the original version of ax-11 1597, before it was discovered (in Jan. 2007) that the shorter ax-11 1597 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 1863.
|- (-. A.x x = y -> (x = y -> (ph -> A.x(x = y -> ph))))
Theorem	ax11 1865	Rederivation of axiom ax-11 1597 from the orginal version, ax-11o 1864. See theorem ax11o 1863 for the derivation of ax-11o 1864 from ax-11 1597. This theorem should not be referenced in any proof. Instead, use ax-11 1597 above so that uses of ax-11 1597 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 1866	A bidirectional version of ax-11o 1864.
|- ((-. A.x x = y /\ x = y) -> (ph <-> A.x(x = y -> ph)))
Theorem	equs5 1867	Lemma used in proofs of substitution properties.
|- (-. A.x x = y -> (E.x(x = y /\ ph) -> A.x(x = y -> ph)))
Theorem	sb3 1868	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 1869	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 1870	Simplified definition of substitution when variables are distinct.
|- (-. A.x x = y -> ([y / x]ph <-> A.x(x = y -> ph)))
Theorem	dfsb2 1871	An alternate definition of proper substitution that, like df-sb 1816, mixes free and bound variables to avoid distinct variable requirements.
|- ([y / x]ph <-> ((x = y /\ ph) \/ A.x(x = y -> ph)))
Theorem	dfsb3 1872	An alternate definition of proper substitution df-sb 1816 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 1873	Bound-variable hypothesis builder for substitution.
|- (-. A.x x = y -> ([y / x]ph -> A.x[y / x]ph))
Theorem	sbequi 1874	An equality theorem for substitution.
|- (x = y -> ([x / z]ph -> [y / z]ph))
Theorem	sbequ 1875	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 1876	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))
Theorem	sbn 1877	Negation inside and outside of substitution are equivalent.
|- ([y / x] -. ph <-> -. [y / x]ph)
Theorem	sbi1 1878	Removal of implication from substitution.
|- ([y / x](ph -> ps) -> ([y / x]ph -> [y / x]ps))
Theorem	sbi2 1879	Introduction of implication into substitution.
|- (([y / x]ph -> [y / x]ps) -> [y / x](ph -> ps))
Theorem	sbim 1880	Implication inside and outside of substitution are equivalent.
|- ([y / x](ph -> ps) <-> ([y / x]ph -> [y / x]ps))
Theorem	sbor 1881	Logical OR inside and outside of substitution are equivalent.
|- ([y / x](ph \/ ps) <-> ([y / x]ph \/ [y / x]ps))
Theorem	sb19.21 1882	Substitution with a variable not free in antecedent affects only the consequent.
|- (ph -> A.xph)   =>   |- ([y / x](ph -> ps) <-> (ph -> [y / x]ps))
Theorem	sban 1883	Conjunction inside and outside of a substitution are equivalent.
|- ([y / x](ph /\ ps) <-> ([y / x]ph /\ [y / x]ps))
Theorem	sb3an 1884	Conjunction inside and outside of a substitution are equivalent.
|- ([y / x](ph /\ ps /\ ch) <-> ([y / x]ph /\ [y / x]ps /\ [y / x]ch))
Theorem	sbbi 1885	Equivalence inside and outside of a substitution are equivalent.
|- ([y / x](ph <-> ps) <-> ([y / x]ph <-> [y / x]ps))
Theorem	sblbis 1886	Introduce left biconditional inside of a substitution.
|- ([y / x]ph <-> ps)   =>   |- ([y / x](ch <-> ph) <-> ([y / x]ch <-> ps))
Theorem	sbrbis 1887	Introduce right biconditional inside of a substitution.
|- ([y / x]ph <-> ps)   =>   |- ([y / x](ph <-> ch) <-> (ps <-> [y / x]ch))
Theorem	sbrbif 1888	Introduce right biconditional inside of a substitution.
|- (ch -> A.xch)   &   |- ([y / x]ph <-> ps)   =>   |- ([y / x](ph <-> ch) <-> (ps <-> ch))
Theorem	a4sbe 1889	A specialization theorem.
|- ([y / x]ph -> E.xph)
Theorem	a4sbim 1890	Specialization of implication. (The proof was shortened by Andrew Salmon, 25-May-2011.)
|- (A.x(ph -> ps) -> ([y / x]ph -> [y / x]ps))
Theorem	a4sbbi 1891	Specialization of biconditional.
|- (A.x(ph <-> ps) -> ([y / x]ph <-> [y / x]ps))
Theorem	sbbid 1892	Deduction substituting both sides of a biconditional.
|- (ph -> A.xph)   &   |- (ph -> (ps <-> ch))   =>   |- (ph -> ([y / x]ps <-> [y / x]ch))
Theorem	sbequ8 1893	Elimination of equality from antecedent after substitution.
|- ([y / x]ph <-> [y / x](x = y -> ph))
Theorem	sbf3t 1894	Substitution has no effect on a non-free variable.
|- (A.x(ph -> A.xph) -> ([y / x]ph <-> ph))
Theorem	hbsb4 1895	A variable not free remains so after substitution with a distinct variable.
|- (ph -> A.zph)   =>   |- (-. A.z z = y -> ([y / x]ph -> A.z[y / x]ph))
Theorem	hbsb4t 1896	A variable not free remains so after substitution with a distinct variable (closed form of hbsb4 1895). (The proof was shortened by Andrew Salmon, 25-May-2011.)
|- (A.xA.z(ph -> A.zph) -> (-. A.z z = y -> ([y / x]ph -> A.z[y / x]ph)))
Theorem	dvelimf 1897	Version of dvelim 2007 without any variable restrictions.
|- (ph -> A.xph)   &   |- (ps -> A.zps)   &   |- (z = y -> (ph <-> ps))   =>   |- (-. A.x x = y -> (ps -> A.xps))
Theorem	dvelimdf 1898	Deduction form of dvelimf 1897. This version may be useful if we want to avoid ax-17 1605 and use ax-16 1854 instead.
|- (ph -> A.xph)   &   |- (ph -> A.zph)   &   |- (ph -> (ps -> A.xps))   &   |- (ph -> (ch -> A.zch))   &   |- (ph -> (z = y -> (ps <-> ch)))   =>   |- (ph -> (-. A.x x = y -> (ch -> A.xch)))
Theorem	sbco 1899	A composition law for substitution.
|- ([y / x][x / y]ph <-> [y / x]ph)
Theorem	sbid2 1900	An identity law for substitution.
|- (ph -> A.xph)   =>   |- ([y / x][x / y]ph <-> ph)