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Statement List for Metamath Proof Explorer - 1801-1900 - Page 19 of 191
TypeLabelDescription
Statement
 
Theorema4im 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)
 
Theorema4ime 1802 Existential introduction, using implicit substitition. Compare Lemma 14 of [Tarski] p. 70.
|- (ph -> A.xph)   &   |- (x = y -> (ph -> ps))   =>   |- (ph -> E.xps)
 
Theorema4imed 1803 Deduction version of a4ime 1802.
|- (ch -> A.xch)   &   |- (ch -> (ph -> A.xph))   &   |- (x = y -> (ph -> ps))   =>   |- (ch -> (ph -> E.xps))
 
Theoremcbv1 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))
 
Theoremcbv2 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))
 
Theoremcbv3 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)
 
Theoremcbv3ALT 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)
 
Theoremcbval 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)
 
Theoremcbvex 1809 Rule used to change bound variables, using implicit substitition.
|- (ph -> A.yph)   &   |- (ps -> A.xps)   &   |- (x = y -> (ph <-> ps))   =>   |- (E.xph <-> E.yps)
 
Theoremchvar 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
 
Theoremequvini 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))
 
Theoremequveli 1812 A variable elimination law for equality with no distinct variable requirements. (Compare equvini 1811.)
|- (A.z(z = x <-> z = y) -> x = y)
 
Theoremhbequid2 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)
 
Syntaxwsbc 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
 
Theoremwsb 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
 
Definitiondf-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)))
 
Theoremsbimi 1817 Infer substitution into antecedent and consequent of an implication.
|- (ph -> ps)   =>   |- ([y / x]ph -> [y / x]ps)
 
Theoremsbbii 1818 Infer substitution into both sides of a logical equivalence.
|- (ph <-> ps)   =>   |- ([y / x]ph <-> [y / x]ps)
 
Theoremdrsb1 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))
 
Theoremsb1 1820 One direction of a simplified definition of substitution.
|- ([y / x]ph -> E.x(x = y /\ ph))
 
Theoremsb2 1821 One direction of a simplified definition of substitution.
|- (A.x(x = y -> ph) -> [y / x]ph)
 
Theoremsbequ1 1822 An equality theorem for substitution.
|- (x = y -> (ph -> [y / x]ph))
 
Theoremsbequ2 1823 An equality theorem for substitution.
|- (x = y -> ([y / x]ph -> ph))
 
Theoremstdpc7 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))
 
Theoremsbequ12 1825 An equality theorem for substitution.
|- (x = y -> (ph <-> [y / x]ph))
 
Theoremsbequ12r 1826 An equality theorem for substitution. (The proof was shortened by Andrew Salmon, 21-Jun-2011.)
|- (x = y -> ([x / y]ph <-> ph))
 
Theoremsbequ12a 1827 An equality theorem for substitution.
|- (x = y -> ([y / x]ph <-> [x / y]ph))
 
Theoremsbid 1828 An identity theorem for substitution. Remark 9.1 in [Megill] p. 447 (p. 15 of the preprint).
|- ([x / x]ph <-> ph)
 
Theoremstdpc4 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)
 
Theoremsbf 1830 Substitution for a variable not free in a wff does not affect it.
|- (ph -> A.xph)   =>   |- ([y / x]ph <-> ph)
 
Theoremsbf2 1831 Substitution has no effect on a bound variable.
|- ([y / x]A.xph <-> A.xph)
 
Theoremsb6x 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))
 
Theoremhbs1f 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)
 
Theoremsbequ5 1834 Substitution does not change an identical variable specifier.
|- ([w / z]A.x x = y <-> A.x x = y)
 
Theoremsbequ6 1835 Substitution does not change a distinctor.
|- ([w / z] -. A.x x = y <-> -. A.x x = y)
 
Theoremsbt 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
 
Theoremequsb1 1837 Substitution applied to an atomic wff.
|- [y / x]x = y
 
Theoremequsb2 1838 Substitution applied to an atomic wff.
|- [y / x]y = x
 
Theoremsbied 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))
 
Theoremsbie 1840 Conversion of implicit substitution to explicit substitution.
|- (ps -> A.xps)   &   |- (x = y -> (ph <-> ps))   =>   |- ([y / x]ph <-> ps)
 
Theorems using axiom ax-11
 
Theoremequs5a 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))
 
Theoremequs5e 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))
 
Theoremsb4a 1843 A version of sb4 1869 that doesn't require a distinctor antecedent.
|- ([y / x]A.yph -> A.x(x = y -> ph))
 
Theoremequs45f 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))
 
Theoremsb6f 1845 Equivalence for substitution when y is not free in ph.
|- (ph -> A.yph)   =>   |- ([y / x]ph <-> A.x(x = y -> ph))
 
Theoremsb5f 1846 Equivalence for substitution when y is not free in ph.
|- (ph -> A.yph)   =>   |- ([y / x]ph <-> E.x(x = y /\ ph))
 
Theoremsb4e 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))
 
Theoremhbsb2a 1848 Special case of a bound-variable hypothesis builder for substitution.
|- ([y / x]A.yph -> A.x[y / x]ph)
 
Theoremhbsb2e 1849 Special case of a bound-variable hypothesis builder for substitution.
|- ([y / x]ph -> A.x[y / x]E.yph)
 
Theoremhbsb3 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
 
Theorema4imv 1851 A version of a4im 1801 with a distinct variable requirement instead of a bound variable hypothesis.
|- (x = y -> (ph -> ps))   =>   |- (A.xph -> ps)
 
Theoremaev 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
 
Theoremax16 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))
 
Axiomax-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))
 
Theoremax17eq 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)
 
Theoremdveeq2 1856 Quantifier introduction when one pair of variables is distinct.
|- (-. A.x x = y -> (z = y -> A.x z = y))
 
Theoremdveeq2ALT 1857 Version of dveeq2 1856 using ax-16 1854 instead of ax-17 1605.
|- (-. A.x x = y -> (z = y -> A.x z = y))
 
TheoremdvelimfALT2 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))
 
Theoremnd5 1859 A lemma for proving conditionless ZFC axioms.
|- (-. A.y y = x -> (z = y -> A.x z = y))
 
Theorem19.23adv 1860 Deduction from Theorem 19.23 of [Margaris] p. 90.
|- (ph -> (ps -> ch))   =>   |- (ph -> (E.xps -> ch))
 
Theoremax11v2 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))))
 
Theoremax11a2 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
 
Theoremax11o 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))))
 
Axiomax-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))))
 
Theoremax11 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
 
Theoremax11b 1866 A bidirectional version of ax-11o 1864.
|- ((-. A.x x = y /\ x = y) -> (ph <-> A.x(x = y -> ph)))
 
Theoremequs5 1867 Lemma used in proofs of substitution properties.
|- (-. A.x x = y -> (E.x(x = y /\ ph) -> A.x(x = y -> ph)))
 
Theoremsb3 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))
 
Theoremsb4 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)))
 
Theoremsb4b 1870 Simplified definition of substitution when variables are distinct.
|- (-. A.x x = y -> ([y / x]ph <-> A.x(x = y -> ph)))
 
Theoremdfsb2 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)))
 
Theoremdfsb3 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)))
 
Theoremhbsb2 1873 Bound-variable hypothesis builder for substitution.
|- (-. A.x x = y -> ([y / x]ph -> A.x[y / x]ph))
 
Theoremsbequi 1874 An equality theorem for substitution.
|- (x = y -> ([x / z]ph -> [y / z]ph))
 
Theoremsbequ 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))
 
Theoremdrsb2 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))
 
Theoremsbn 1877 Negation inside and outside of substitution are equivalent.
|- ([y / x] -. ph <-> -. [y / x]ph)
 
Theoremsbi1 1878 Removal of implication from substitution.
|- ([y / x](ph -> ps) -> ([y / x]ph -> [y / x]ps))
 
Theoremsbi2 1879 Introduction of implication into substitution.
|- (([y / x]ph -> [y / x]ps) -> [y / x](ph -> ps))
 
Theoremsbim 1880 Implication inside and outside of substitution are equivalent.
|- ([y / x](ph -> ps) <-> ([y / x]ph -> [y / x]ps))
 
Theoremsbor 1881 Logical OR inside and outside of substitution are equivalent.
|- ([y / x](ph \/ ps) <-> ([y / x]ph \/ [y / x]ps))
 
Theoremsb19.21 1882 Substitution with a variable not free in antecedent affects only the consequent.
|- (ph -> A.xph)   =>   |- ([y / x](ph -> ps) <-> (ph -> [y / x]ps))
 
Theoremsban 1883 Conjunction inside and outside of a substitution are equivalent.
|- ([y / x](ph /\ ps) <-> ([y / x]ph /\ [y / x]ps))
 
Theoremsb3an 1884 Conjunction inside and outside of a substitution are equivalent.
|- ([y / x](ph /\ ps /\ ch) <-> ([y / x]ph /\ [y / x]ps /\ [y / x]ch))
 
Theoremsbbi 1885 Equivalence inside and outside of a substitution are equivalent.
|- ([y / x](ph <-> ps) <-> ([y / x]ph <-> [y / x]ps))
 
Theoremsblbis 1886 Introduce left biconditional inside of a substitution.
|- ([y / x]ph <-> ps)   =>   |- ([y / x](ch <-> ph) <-> ([y / x]ch <-> ps))
 
Theoremsbrbis 1887 Introduce right biconditional inside of a substitution.
|- ([y / x]ph <-> ps)   =>   |- ([y / x](ph <-> ch) <-> (ps <-> [y / x]ch))
 
Theoremsbrbif 1888 Introduce right biconditional inside of a substitution.
|- (ch -> A.xch)   &   |- ([y / x]ph <-> ps)   =>   |- ([y / x](ph <-> ch) <-> (ps <-> ch))
 
Theorema4sbe 1889 A specialization theorem.
|- ([y / x]ph -> E.xph)
 
Theorema4sbim 1890 Specialization of implication. (The proof was shortened by Andrew Salmon, 25-May-2011.)
|- (A.x(ph -> ps) -> ([y / x]ph -> [y / x]ps))
 
Theorema4sbbi 1891 Specialization of biconditional.
|- (A.x(ph <-> ps) -> ([y / x]ph <-> [y / x]ps))
 
Theoremsbbid 1892 Deduction substituting both sides of a biconditional.
|- (ph -> A.xph)   &   |- (ph -> (ps <-> ch))   =>   |- (ph -> ([y / x]ps <-> [y / x]ch))
 
Theoremsbequ8 1893 Elimination of equality from antecedent after substitution.
|- ([y / x]ph <-> [y / x](x = y -> ph))
 
Theoremsbf3t 1894 Substitution has no effect on a non-free variable.
|- (A.x(ph -> A.xph) -> ([y / x]ph <-> ph))
 
Theoremhbsb4 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))
 
Theoremhbsb4t 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)))
 
Theoremdvelimf 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))
 
Theoremdvelimdf 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)))
 
Theoremsbco 1899 A composition law for substitution.
|- ([y / x][x / y]ph <-> [y / x]ph)
 
Theoremsbid2 1900 An identity law for substitution.
|- (ph -> A.xph)   =>   |- ([y / x][x / y]ph <-> ph)

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