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Theorem List for Metamath Proof Explorer - 1801-1900   *Has distinct variable group(s)
TypeLabelDescription
Statement

Theoremexcom13 1801 Swap 1st and 3rd existential quantifiers. (Contributed by NM, 9-Mar-1995.)

Theoremexrot3 1802 Rotate existential quantifiers. (Contributed by NM, 17-Mar-1995.)

Theoremexrot4 1803 Rotate existential quantifiers twice. (Contributed by NM, 9-Mar-1995.)

Theoremnexr 1804 Inference from 19.8a 1758. (Contributed by Jeff Hankins, 26-Jul-2009.)

Theoremnfim1 1805 A closed form of nfim 1735. (Contributed by NM, 5-Aug-1993.) (Revised by Mario Carneiro, 24-Sep-2016.)

Theoremnfan1 1806 A closed form of nfan 1737. (Contributed by Mario Carneiro, 3-Oct-2016.)

Theoremexan 1807 Place a conjunct in the scope of an existential quantifier. (Contributed by NM, 18-Aug-1993.) (Proof shortened by Andrew Salmon, 25-May-2011.)

Theoremhbnd 1808 Deduction form of bound-variable hypothesis builder hbn 1722. (Contributed by NM, 3-Jan-2002.)

Theoremhbimd 1809 Deduction form of bound-variable hypothesis builder hbim 1723. (Contributed by NM, 1-Jan-2002.)

Theoremhbim1 1810 A closed form of hbim 1723. (Contributed by NM, 5-Aug-1993.)

Theoremaaan 1811 Rearrange universal quantifiers. (Contributed by NM, 12-Aug-1993.)

Theoremeeor 1812 Rearrange existential quantifiers. (Contributed by NM, 8-Aug-1994.)

Theoremqexmid 1813 Quantified "excluded middle." Exercise 9.2a of Boolos, p. 111, Computability and Logic. (Contributed by NM, 10-Dec-2000.)

1.5.11  Equality theorems without distinct variables

Theoremax9o 1814 Show that the original axiom ax-9o 1815 can be derived from ax-9 1684 and others. See ax9from9o 1816 for the rederivation of ax-9 1684 from ax-9o 1815.

Normally, ax9o 1814 should be used rather than ax-9o 1815, except by theorems specifically studying the latter's properties. (Contributed by NM, 5-Aug-1993.) (Proof modification is discouraged.)

Axiomax-9o 1815 A variant of ax-9 1684. Axiom scheme C10' in [Megill] p. 448 (p. 16 of the preprint).

This axiom is redundant, as shown by theorem ax9o 1814.

Normally, ax9o 1814 should be used rather than ax-9o 1815, except by theorems specifically studying the latter's properties. (Contributed by NM, 5-Aug-1993.) (New usage is discouraged.)

Theoremax9from9o 1816 Rederivation of axiom ax-9 1684 from the orginal version, ax-9o 1815. See ax9o 1814 for the derivation of ax-9o 1815 from ax-9 1684. Lemma L18 in [Megill] p. 446 (p. 14 of the preprint).

This theorem should not be referenced in any proof. Instead, use ax-9 1684 above so that uses of ax-9 1684 can be more easily identified. (Contributed by NM, 5-Aug-1993.) (Proof modification is discouraged.) (New usage is discouraged.)

Theorema9e 1817 At least one individual exists. This is not a theorem of free logic, which is sound in empty domains. For such a logic, we would add this theorem as an axiom of set theory (Axiom 0 of [Kunen] p. 10). In the system consisting of ax-5 1533 through ax-14 1626 and ax-17 1628, all axioms other than ax-9 1684 are believed to be theorems of free logic, although the system without ax-9 1684 is probably not complete in free logic. (Contributed by NM, 5-Aug-1993.)

Theoremequid 1818 Identity law for equality (reflexivity). Lemma 6 of [Tarski] p. 68. This is often an axiom of equality in textbook systems, but we don't need it as an axiom since it can be proved from our other axioms (although the proof, as you can see below, is not as obvious as you might think). This proof uses only axioms without distinct variable conditions and thus requires no dummy variables. A simpler proof, similar to Tarki's, is possible if we make use of ax-17 1628; see the proof of equid1 1820. See equidALT 1819 for an alternate proof. (Contributed by NM, 30-Nov-2008.) (Proof modification is discouraged.)

TheoremequidALT 1819 Identity law for equality (reflexivity). Lemma 6 of [Tarski] p. 68. Alternate proof of equid 1818 from older axioms ax-6o 1697 and ax-9o 1815. (Contributed by NM, 5-Aug-1993.) (Proof modification is discouraged.) (New usage is discouraged.)

Theoremequid1 1820 Identity law for equality (reflexivity). Lemma 6 of [Tarski] p. 68. This proof is similar to Tarski's and makes use of a dummy variable . See equid 1818 for a proof that avoids dummy variables (but is less intuitive). (Contributed by NM, 1-Apr-2005.) (Proof modification is discouraged.)

Theoremstdpc6 1821 One of the two equality axioms of standard predicate calculus, called reflexivity of equality. (The other one is stdpc7 1891.) 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.)

Theoremequcomi 1822 Commutative law for equality. Lemma 7 of [Tarski] p. 69. (Contributed by NM, 5-Aug-1993.) (Revised by NM, 28-Nov-2013.)

Theoremequcomi-o 1823 Commutative law for equality. Lemma 7 of [Tarski] p. 69. Version of equcomi 1822 not requiring ax-17 1628. (Contributed by NM, 5-Aug-1993.)

Theoremequcom 1824 Commutative law for equality. (Contributed by NM, 20-Aug-1993.)

Theoremequcoms 1825 An inference commuting equality in antecedent. Used to eliminate the need for a syllogism. (Contributed by NM, 5-Aug-1993.)

Theoremequtr 1826 A transitive law for equality. (Contributed by NM, 23-Aug-1993.)

Theoremequtrr 1827 A transitive law for equality. Lemma L17 in [Megill] p. 446 (p. 14 of the preprint). (Contributed by NM, 23-Aug-1993.)

Theoremequtr2 1828 A transitive law for equality. (Contributed by NM, 12-Aug-1993.) (Proof shortened by Andrew Salmon, 25-May-2011.)

Theoremequequ1 1829 An equivalence law for equality. (Contributed by NM, 5-Aug-1993.)

Theoremequequ2 1830 An equivalence law for equality. (Contributed by NM, 5-Aug-1993.)

Theoremelequ1 1831 An identity law for the non-logical predicate. (Contributed by NM, 5-Aug-1993.)

Theoremelequ2 1832 An identity law for the non-logical predicate. (Contributed by NM, 5-Aug-1993.)

Theoremax11i 1833 Inference that has ax-11 1624 (without ) as its conclusion and doesn't require ax-10 1678, ax-11 1624, or ax-12o 1664 for its proof. 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.)

1.5.12  Axioms ax-10 and ax-11

Theoremax10o 1834 Show that ax-10o 1835 can be derived from ax-10 1678. An open problem is whether this theorem can be derived from ax-10 1678 and the others when ax-11 1624 is replaced with ax-11o 1940. See theorem ax10from10o 1836 for the rederivation of ax-10 1678 from ax10o 1834.

Normally, ax10o 1834 should be used rather than ax-10o 1835, except by theorems specifically studying the latter's properties. (Contributed by NM, 16-May-2008.)

Axiomax-10o 1835 Axiom ax-10o 1835 ("o" for "old") was the original version of ax-10 1678, before it was discovered (in May 2008) that the shorter ax-10 1678 could replace it. It appears as Axiom scheme C11' in [Megill] p. 448 (p. 16 of the preprint).

This axiom is redundant, as shown by theorem ax10o 1834.

Normally, ax10o 1834 should be used rather than ax-10o 1835, except by theorems specifically studying the latter's properties. (Contributed by NM, 5-Aug-1993.) (New usage is discouraged.)

Theoremax10from10o 1836 Rederivation of ax-10 1678 from original version ax-10o 1835. See theorem ax10o 1834 for the derivation of ax-10o 1835 from ax-10 1678.

This theorem should not be referenced in any proof. Instead, use ax-10 1678 above so that uses of ax-10 1678 can be more easily identified, or use alequcom-o 1837 when this form is needed for studies involving ax-10o 1835 and omitting ax-17 1628. (Contributed by NM, 16-May-2008.) (Proof modification is discouraged.) (New usage is discouraged.)

Theoremalequcom-o 1837 Commutation law for identical variable specifiers. The antecedent and consequent are true when and are substituted with the same variable. Lemma L12 in [Megill] p. 445 (p. 12 of the preprint). Version of alequcom 1680 using ax-10o 1835. Unlike ax10from10o 1836, this version does not require ax-17 1628. (Contributed by NM, 5-Aug-1993.)

Theoremalequcoms-o 1838 A commutation rule for identical variable specifiers. Version of alequcoms 1681 using ax-10o . (Contributed by NM, 5-Aug-1993.)

Theoremnalequcoms-o 1839 A commutation rule for distinct variable specifiers. Version of nalequcoms 1682 using ax-10o 1835. (Contributed by NM, 2-Jan-2002.)

Theoremhbae 1840 All variables are effectively bound in an identical variable specifier. (Contributed by NM, 5-Aug-1993.)

Theoremhbae-o 1841 All variables are effectively bound in an identical variable specifier. Version of hbae 1840 using ax-10o 1835. (Contributed by NM, 5-Aug-1993.) (Proof modification is disccouraged.) (New usage is discouraged.)

Theoremnfae 1842 All variables are effectively bound in an identical variable specifier. (Contributed by Mario Carneiro, 11-Aug-2016.)

Theoremhbaes 1843 Rule that applies hbae 1840 to antecedent. (Contributed by NM, 5-Aug-1993.)

Theoremhbnae 1844 All variables are effectively bound in a distinct variable specifier. Lemma L19 in [Megill] p. 446 (p. 14 of the preprint). (Contributed by NM, 5-Aug-1993.)

Theoremhbnae-o 1845 All variables are effectively bound in a distinct variable specifier. Lemma L19 in [Megill] p. 446 (p. 14 of the preprint). Version of hbnae 1844 using ax-10o 1835. (Contributed by NM, 5-Aug-1993.)

Theoremnfnae 1846 All variables are effectively bound in an distinct variable specifier. (Contributed by Mario Carneiro, 11-Aug-2016.)

Theoremhbnaes 1847 Rule that applies hbnae 1844 to antecedent. (Contributed by NM, 5-Aug-1993.)

Theoremnfeqf 1848 A variable is effectively not free in an equality if it is not either of the involved variables. version of ax-12o 1664. (Contributed by Mario Carneiro, 6-Oct-2016.)

Theoremequs4 1849 Lemma used in proofs of substitution properties. (Contributed by NM, 5-Aug-1993.) (Proof shortened by Mario Carneiro, 20-May-2014.)

Theoremequsal 1850 A useful equivalence related to substitution. (Contributed by NM, 5-Aug-1993.) (Revised by Mario Carneiro, 3-Oct-2016.)

Theoremequsalh 1851 A useful equivalence related to substitution. (Contributed by NM, 5-Aug-1993.) (Proof shortened by Andrew Salmon, 12-Aug-2011.)

Theoremequsex 1852 A useful equivalence related to substitution. (Contributed by NM, 5-Aug-1993.) (Revised by Mario Carneiro, 3-Oct-2016.)

TheoremdvelimfALT 1853 Proof of dvelimh 1974 that uses ax-10o 1835 (in the form of ax10o 1834) but not ax-11o 1940, ax-10 1678, or ax-11 1624 (if we replace uses of ax10o 1834 by ax-10o 1835 in the proofs of referenced theorems). See dvelimALT 2094 for a proof (of the distinct variable version dvelim 2092) that doesn't require ax-10 1678. It is not clear whether a proof is possible that uses ax-10 1678 but avoids ax-11 1624, ax-11o 1940, and ax-10o 1835. (Contributed by NM, 12-Nov-2002.) (Proof modification is discouraged.)

Theoremdvelimf-o 1854 Proof of dvelimh 1974 that uses ax-10o 1835 but not ax-11o 1940, ax-10 1678, or ax-11 1624. Version of dvelimfALT 1853 using ax-10o 1835 instead of ax10o 1834. (Contributed by NM, 12-Nov-2002.)

Theoremdral1 1855 Formula-building lemma for use with the Distinctor Reduction Theorem. Part of Theorem 9.4 of [Megill] p. 448 (p. 16 of preprint). (Contributed by NM, 24-Nov-1994.)

Theoremdral1-o 1856 Formula-building lemma for use with the Distinctor Reduction Theorem. Part of Theorem 9.4 of [Megill] p. 448 (p. 16 of preprint). Version of dral1 1855 using ax-10o 1835. (Contributed by NM, 24-Nov-1994.)

Theoremdral2 1857 Formula-building lemma for use with the Distinctor Reduction Theorem. Part of Theorem 9.4 of [Megill] p. 448 (p. 16 of preprint). (Contributed by NM, 27-Feb-2005.)

Theoremdral2-o 1858 Formula-building lemma for use with the Distinctor Reduction Theorem. Part of Theorem 9.4 of [Megill] p. 448 (p. 16 of preprint). Version of dral2 1857 using ax-10o 1835. (Contributed by NM, 27-Feb-2005.)

Theoremdrex1 1859 Formula-building lemma for use with the Distinctor Reduction Theorem. Part of Theorem 9.4 of [Megill] p. 448 (p. 16 of preprint). (Contributed by NM, 27-Feb-2005.)

Theoremdrex2 1860 Formula-building lemma for use with the Distinctor Reduction Theorem. Part of Theorem 9.4 of [Megill] p. 448 (p. 16 of preprint). (Contributed by NM, 27-Feb-2005.)

Theoremdrnf1 1861 Formula-building lemma for use with the Distinctor Reduction Theorem. (Contributed by Mario Carneiro, 4-Oct-2016.)

Theoremdrnf2 1862 Formula-building lemma for use with the Distinctor Reduction Theorem. (Contributed by Mario Carneiro, 4-Oct-2016.)

Theoremexdistrf 1863 Distribution of existential quantifiers, with a bound-variable hypothesis saying that is not free in , but can be free in (and there is no distinct variable condition on and ). (Contributed by Mario Carneiro, 20-Mar-2013.)

Theoremnfald2 1864 Variation on nfald 1742 which adds the hypothesis that and are distinct in the inner subproof. (Contributed by Mario Carneiro, 8-Oct-2016.)

Theoremnfexd2 1865 Variation on nfexd 1743 which adds the hypothesis that and are distinct in the inner subproof. (Contributed by Mario Carneiro, 8-Oct-2016.)

Theorema4imt 1866 Closed theorem form of a4im 1867. (Contributed by NM, 15-Jan-2008.) (Revised by Mario Carneiro, 17-Oct-2016.)

Theorema4im 1867 Specialization, using implicit substitution. Compare Lemma 14 of [Tarski] p. 70. The a4im 1867 series of theorems requires that only one direction of the substitution hypothesis hold. (Contributed by NM, 5-Aug-1993.) (Revised by Mario Carneiro, 3-Oct-2016.)

Theorema4ime 1868 Existential introduction, using implicit substitution. Compare Lemma 14 of [Tarski] p. 70. (Contributed by NM, 7-Aug-1994.) (Revised by Mario Carneiro, 3-Oct-2016.)

Theorema4imed 1869 Deduction version of a4ime 1868. (Contributed by NM, 5-Aug-1993.) (Revised by Mario Carneiro, 3-Oct-2016.)

Theoremcbv1h 1870 Rule used to change bound variables, using implicit substitution. (Contributed by NM, 5-Aug-1993.)

Theoremcbv1 1871 Rule used to change bound variables, using implicit substitution. (Contributed by NM, 5-Aug-1993.) (Revised by Mario Carneiro, 3-Oct-2016.)

Theoremcbv2h 1872 Rule used to change bound variables, using implicit substitution. (Contributed by NM, 5-Aug-1993.)

Theoremcbv2 1873 Rule used to change bound variables, using implicit substitution. (Contributed by NM, 5-Aug-1993.) (Revised by Mario Carneiro, 3-Oct-2016.)

Theoremcbv3 1874 Rule used to change bound variables, using implicit substitution, that does not use ax-12o 1664. (Contributed by NM, 5-Aug-1993.)

Theoremcbv3ALT 1875 Rule used to change bound variables, using implicit substitution. (Contributed by NM, 5-Aug-1993.) (Proof shortened by Andrew Salmon, 25-May-2011.) (Proof modification is discouraged.)

Theoremcbval 1876 Rule used to change bound variables, using implicit substitution. (Contributed by NM, 5-Aug-1993.) (Revised by Mario Carneiro, 3-Oct-2016.)

Theoremcbvex 1877 Rule used to change bound variables, using implicit substitution. (Contributed by NM, 5-Aug-1993.)

Theoremchvar 1878 Implicit substitution of for into a theorem. (Contributed by Raph Levien, 9-Jul-2003.) (Revised by Mario Carneiro, 3-Oct-2016.)

Theoremequvini 1879 A variable introduction law for equality. Lemma 15 of [Monk2] p. 109, however we do not require to be distinct from and (making the proof longer). (Contributed by NM, 5-Aug-1993.) (Proof shortened by Andrew Salmon, 25-May-2011.)

Theoremequveli 1880 A variable elimination law for equality with no distinct variable requirements. (Compare equvini 1879.) (Contributed by NM, 1-Mar-2013.) (Proof shortened by Mario Carneiro, 17-Oct-2016.)

Theoremax12 1881 Derive ax-12 1633 from ax-12o 1664. (Contributed by NM, 21-Dec-2015.) (New usage is discouraged.)

1.5.13  Substitution (without distinct variables)

Syntaxwsb 1882 Extend wff definition to include proper substitution (read "the wff that results when is properly substituted for in wff "). (Contributed by NM, 24-Jan-2006.)

Definitiondf-sb 1883 Define proper substitution. Remark 9.1 in [Megill] p. 447 (p. 15 of the preprint). For our notation, we use to mean "the wff that results from the proper substitution of for in the wff ." We can also use in place of the "free for" side condition used in traditional predicate calculus; see, for example, stdpc4 1896.

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, " is the wff that results when is properly substituted for in ." For example, if the original is , then is , from which we obtain that is . So what exactly does 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 1952, sbcom2 2074 and sbid2v 2083).

Note that our definition is valid even when and are replaced with the same variable, as sbid 1895 shows. We achieve this by having 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 2079 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 1948. When and are distinct, we can express proper substitution with the simpler expressions of sb5 1993 and sb6 1992.

There are no restrictions on any of the variables, including what variables may occur in wff . (Contributed by NM, 5-Aug-1993.)

Theoremsbimi 1884 Infer substitution into antecedent and consequent of an implication. (Contributed by NM, 25-Jun-1998.)

Theoremsbbii 1885 Infer substitution into both sides of a logical equivalence. (Contributed by NM, 5-Aug-1993.)

Theoremdrsb1 1886 Formula-building lemma for use with the Distinctor Reduction Theorem. Part of Theorem 9.4 of [Megill] p. 448 (p. 16 of preprint). (Contributed by NM, 5-Aug-1993.)

Theoremsb1 1887 One direction of a simplified definition of substitution. (Contributed by NM, 5-Aug-1993.)

Theoremsb2 1888 One direction of a simplified definition of substitution. (Contributed by NM, 5-Aug-1993.)

Theoremsbequ1 1889 An equality theorem for substitution. (Contributed by NM, 5-Aug-1993.)

Theoremsbequ2 1890 An equality theorem for substitution. (Contributed by NM, 5-Aug-1993.)

Theoremstdpc7 1891 One of the two equality axioms of standard predicate calculus, called substitutivity of equality. (The other one is stdpc6 1821.) Translated to traditional notation, it can be read: " , provided that is free for in ." Axiom 7 of [Mendelson] p. 95. (Contributed by NM, 15-Feb-2005.)

Theoremsbequ12 1892 An equality theorem for substitution. (Contributed by NM, 5-Aug-1993.)

Theoremsbequ12r 1893 An equality theorem for substitution. (Contributed by NM, 6-Oct-2004.) (Proof shortened by Andrew Salmon, 21-Jun-2011.)

Theoremsbequ12a 1894 An equality theorem for substitution. (Contributed by NM, 5-Aug-1993.)

Theoremsbid 1895 An identity theorem for substitution. Remark 9.1 in [Megill] p. 447 (p. 15 of the preprint). (Contributed by NM, 5-Aug-1993.)

Theoremstdpc4 1896 The specialization axiom of standard predicate calculus. It states that if a statement holds for all , then it also holds for the specific case of (properly) substituted for . Translated to traditional notation, it can be read: " , provided that is free for in ." Axiom 4 of [Mendelson] p. 69. See also a4sbc 2933 and ra4sbc 2999. (Contributed by NM, 5-Aug-1993.)

Theoremsbft 1897 Substitution has no effect on a non-free variable. (Contributed by NM, 30-May-2009.) (Revised by Mario Carneiro, 12-Oct-2016.)

Theoremsbf 1898 Substitution for a variable not free in a wff does not affect it. (Contributed by NM, 5-Aug-1993.) (Revised by Mario Carneiro, 4-Oct-2016.)

Theoremsbh 1899 Substitution for a variable not free in a wff does not affect it. (Contributed by NM, 5-Aug-1993.)

Theoremsbf2 1900 Substitution has no effect on a bound variable. (Contributed by NM, 1-Jul-2005.)

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