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Theorem List for Metamath Proof Explorer - 1901-2000   *Has distinct variable group(s)
TypeLabelDescription
Statement

Theoremnaecoms 1901 A commutation rule for distinct variable specifiers. (Contributed by NM, 2-Jan-2002.)

Theoremax9 1902 Theorem showing that ax-9 1644 follows from the weaker version ax9v 1645. (Even though this theorem depends on ax-9 1644, all references of ax-9 1644 are made via ax9v 1645. An earlier version stated ax9v 1645 as a separate axiom, but having two axioms caused some confusion.)

This theorem should be referenced in place of ax-9 1644 so that all proofs can be traced back to ax9v 1645. (Contributed by NM, 12-Nov-2013.) (Revised by NM, 25-Jul-2015.)

Theoremax9o 1903 Show that the original axiom ax-9o 2090 can be derived from ax9 1902 and others. See ax9from9o 2100 for the rederivation of ax9 1902 from ax-9o 2090.

Normally, ax9o 1903 should be used rather than ax-9o 2090, except by theorems specifically studying the latter's properties. (Contributed by NM, 5-Aug-1993.) (Proof modification is discouraged.)

Theorema9e 1904 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 1547 through ax-14 1700 and ax-17 1606, all axioms other than ax9 1902 are believed to be theorems of free logic, although the system without ax9 1902 is probably not complete in free logic. (Contributed by NM, 5-Aug-1993.)

Theoremax10o 1905 Show that ax-10o 2091 can be derived from ax-10 2092 in the form of ax10 1897. Normally, ax10o 1905 should be used rather than ax-10o 2091, except by theorems specifically studying the latter's properties. (Contributed by NM, 16-May-2008.) (Proof modification is discouraged.)

Theoremhbae 1906 All variables are effectively bound in an identical variable specifier. (Contributed by NM, 5-Aug-1993.)

Theoremnfae 1907 All variables are effectively bound in an identical variable specifier. (Contributed by Mario Carneiro, 11-Aug-2016.)

Theoremhbnae 1908 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.)

Theoremnfnae 1909 All variables are effectively bound in a distinct variable specifier. (Contributed by Mario Carneiro, 11-Aug-2016.)

Theoremhbnaes 1910 Rule that applies hbnae 1908 to antecedent. (Contributed by NM, 5-Aug-1993.)

Theoremnfeqf 1911 A variable is effectively not free in an equality if it is not either of the involved variables. version of ax-12o 2094. (Contributed by Mario Carneiro, 6-Oct-2016.)

Theoremequs4 1912 Lemma used in proofs of substitution properties. (Contributed by NM, 5-Aug-1993.) (Proof shortened by Mario Carneiro, 20-May-2014.)

Theoremequsal 1913 A useful equivalence related to substitution. (Contributed by NM, 5-Aug-1993.) (Proof shortened by Andrew Salmon, 12-Aug-2011.) (Revised by Mario Carneiro, 3-Oct-2016.)

Theoremequsalh 1914 A useful equivalence related to substitution. (Contributed by NM, 5-Aug-1993.)

Theoremequsex 1915 A useful equivalence related to substitution. (Contributed by NM, 5-Aug-1993.) (Revised by Mario Carneiro, 3-Oct-2016.)

Theoremequsexh 1916 A useful equivalence related to substitution. (Contributed by NM, 5-Aug-1993.)

Theoremdvelimh 1917 Version of dvelim 1969 without any variable restrictions. (Contributed by NM, 1-Oct-2002.)

Theoremdral1 1918 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.)

Theoremdral2 1919 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.)

Theoremdrex1 1920 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 1921 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 1922 Formula-building lemma for use with the Distinctor Reduction Theorem. (Contributed by Mario Carneiro, 4-Oct-2016.)

Theoremdrnf2 1923 Formula-building lemma for use with the Distinctor Reduction Theorem. (Contributed by Mario Carneiro, 4-Oct-2016.)

Theoremexdistrf 1924 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 1925 Variation on nfald 1787 which adds the hypothesis that and are distinct in the inner subproof. (Contributed by Mario Carneiro, 8-Oct-2016.)

Theoremnfexd2 1926 Variation on nfexd 1788 which adds the hypothesis that and are distinct in the inner subproof. (Contributed by Mario Carneiro, 8-Oct-2016.)

Theoremspimt 1927 Closed theorem form of spim 1928. (Contributed by NM, 15-Jan-2008.) (Revised by Mario Carneiro, 17-Oct-2016.)

Theoremspim 1928 Specialization, using implicit substitution. Compare Lemma 14 of [Tarski] p. 70. The spim 1928 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.)

Theoremspime 1929 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.)

Theoremspimed 1930 Deduction version of spime 1929. (Contributed by NM, 5-Aug-1993.) (Revised by Mario Carneiro, 3-Oct-2016.)

Theoremcbv1h 1931 Rule used to change bound variables, using implicit substitution. (Contributed by NM, 5-Aug-1993.)

Theoremcbv1 1932 Rule used to change bound variables, using implicit substitution. (Contributed by NM, 5-Aug-1993.) (Revised by Mario Carneiro, 3-Oct-2016.)

Theoremcbv2h 1933 Rule used to change bound variables, using implicit substitution. (Contributed by NM, 5-Aug-1993.)

Theoremcbv2 1934 Rule used to change bound variables, using implicit substitution. (Contributed by NM, 5-Aug-1993.) (Revised by Mario Carneiro, 3-Oct-2016.)

Theoremcbv3 1935 Rule used to change bound variables, using implicit substitution, that does not use ax-12o 2094. (Contributed by NM, 5-Aug-1993.)

Theoremcbv3h 1936 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 1937 Rule used to change bound variables, using implicit substitution. (Contributed by NM, 5-Aug-1993.) (Revised by Mario Carneiro, 3-Oct-2016.)

Theoremcbvex 1938 Rule used to change bound variables, using implicit substitution. (Contributed by NM, 5-Aug-1993.)

Theoremchvar 1939 Implicit substitution of for into a theorem. (Contributed by Raph Levien, 9-Jul-2003.) (Revised by Mario Carneiro, 3-Oct-2016.)

Theoremequvini 1940 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 1941 A variable elimination law for equality with no distinct variable requirements. (Compare equvini 1940.) (Contributed by NM, 1-Mar-2013.) (Proof shortened by Mario Carneiro, 17-Oct-2016.)

Theoremequs45f 1942 Two ways of expressing substitution when is not free in . (Contributed by NM, 25-Apr-2008.) (Revised by Mario Carneiro, 4-Oct-2016.)

Theoremspimv 1943* A version of spim 1928 with a distinct variable requirement instead of a bound variable hypothesis. (Contributed by NM, 5-Aug-1993.)

Theoremaev 1944* A "distinctor elimination" lemma with no restrictions on variables in the consequent. (Contributed by NM, 8-Nov-2006.)

Theoremax11v2 1945* Recovery of ax-11o 2093 from ax11v 2049. This proof uses ax-10 2092 and ax-11 1727. TODO: figure out if this is useful, or if it should be simplified or eliminated. (Contributed by NM, 2-Feb-2007.)

Theoremax11a2 1946* Derive ax-11o 2093 from a hypothesis in the form of ax-11 1727. ax-10 2092 and ax-11 1727 are used by the proof, but not ax-10o 2091 or ax-11o 2093. TODO: figure out if this is useful, or if it should be simplified or eliminated. (Contributed by NM, 2-Feb-2007.)

Theoremax11o 1947 Derivation of set.mm's original ax-11o 2093 from ax-10 2092 and the shorter ax-11 1727 that has replaced it.

An open problem is whether this theorem can be proved without relying on ax-16 2096 or ax-17 1606 (given all of the original and new versions of sp 1728 through ax-15 2095).

Another open problem is whether this theorem can be proved without relying on ax12o 1887.

Theorem ax11 2107 shows the reverse derivation of ax-11 1727 from ax-11o 2093.

Normally, ax11o 1947 should be used rather than ax-11o 2093, except by theorems specifically studying the latter's properties. (Contributed by NM, 3-Feb-2007.)

Theoremax11b 1948 A bidirectional version of ax11o 1947. (Contributed by NM, 30-Jun-2006.)

Theoremequs5 1949 Lemma used in proofs of substitution properties. (Contributed by NM, 5-Aug-1993.)

Theoremdvelimf 1950 Version of dvelimv 1892 without any variable restrictions. (Contributed by NM, 1-Oct-2002.) (Revised by Mario Carneiro, 6-Oct-2016.)

Theoremspv 1951* Specialization, using implicit substitution. (Contributed by NM, 30-Aug-1993.)

Theoremspimev 1952* Distinct-variable version of spime 1929. (Contributed by NM, 5-Aug-1993.)

Theoremspeiv 1953* Inference from existential specialization, using implicit substitution. (Contributed by NM, 19-Aug-1993.)

Theoremequvin 1954* A variable introduction law for equality. Lemma 15 of [Monk2] p. 109. (Contributed by NM, 5-Aug-1993.)

Theoremcbvalv 1955* Rule used to change bound variables, using implicit substitution. (Contributed by NM, 5-Aug-1993.)

Theoremcbvexv 1956* Rule used to change bound variables, using implicit substitution. (Contributed by NM, 5-Aug-1993.)

Theoremcbval2 1957* Rule used to change bound variables, using implicit substitution. (Contributed by NM, 22-Dec-2003.) (Revised by Mario Carneiro, 6-Oct-2016.)

Theoremcbvex2 1958* Rule used to change bound variables, using implicit substitution. (Contributed by NM, 14-Sep-2003.) (Revised by Mario Carneiro, 6-Oct-2016.)

Theoremcbval2v 1959* Rule used to change bound variables, using implicit substitution. (Contributed by NM, 4-Feb-2005.)

Theoremcbvex2v 1960* Rule used to change bound variables, using implicit substitution. (Contributed by NM, 26-Jul-1995.)

Theoremcbvald 1961* Deduction used to change bound variables, using implicit substitution, particularly useful in conjunction with dvelim 1969. (Contributed by NM, 2-Jan-2002.) (Revised by Mario Carneiro, 6-Oct-2016.)

Theoremcbvexd 1962* Deduction used to change bound variables, using implicit substitution, particularly useful in conjunction with dvelim 1969. (Contributed by NM, 2-Jan-2002.) (Revised by Mario Carneiro, 6-Oct-2016.)

Theoremcbvaldva 1963* Rule used to change the bound variable in a universal quantifier with implicit substitution. Deduction form. (Contributed by David Moews, 1-May-2017.)

Theoremcbvexdva 1964* Rule used to change the bound variable in an existential quantifier with implicit substitution. Deduction form. (Contributed by David Moews, 1-May-2017.)

Theoremcbvex4v 1965* Rule used to change bound variables, using implicit substitution. (Contributed by NM, 26-Jul-1995.)

Theoremchvarv 1966* Implicit substitution of for into a theorem. (Contributed by NM, 20-Apr-1994.)

Theoremcleljust 1967* When the class variables in definition df-clel 2292 are replaced with set variables, this theorem of predicate calculus is the result. This theorem provides part of the justification for the consistency of that definition, which "overloads" the set variables in wel 1697 with the class variables in wcel 1696. Note: This proof is referenced on the Metamath Proof Explorer Home Page and shouldn't be changed. (Contributed by NM, 28-Jan-2004.) (Proof modification is discouraged.)

TheoremcleljustALT 1968* When the class variables in definition df-clel 2292 are replaced with set variables, this theorem of predicate calculus is the result. This theorem provides part of the justification for the consistency of that definition, which "overloads" the set variables in wel 1697 with the class variables in wcel 1696. (Contributed by NM, 28-Jan-2004.) (Revised by Mario Carneiro, 21-Dec-2016.)

Theoremdvelim 1969* This theorem can be used to eliminate a distinct variable restriction on and and replace it with the "distinctor" as an antecedent. normally has free and can be read , and substitutes for and can be read . We don't require that and be distinct: if they aren't, the distinctor will become false (in multiple-element domains of discourse) and "protect" the consequent.

To obtain a closed-theorem form of this inference, prefix the hypotheses with , conjoin them, and apply dvelimdf 2035.

Other variants of this theorem are dvelimh 1917 (with no distinct variable restrictions), dvelimhw 1747 (that avoids ax-12 1878), and dvelimALT 2085 (that avoids ax-10 2092). (Contributed by NM, 23-Nov-1994.)

Theoremdvelimnf 1970* Version of dvelim 1969 using "not free" notation. (Contributed by Mario Carneiro, 9-Oct-2016.)

Theoremdveeq1 1971* Quantifier introduction when one pair of variables is distinct. (Contributed by NM, 2-Jan-2002.)

Theoremdveel1 1972* Quantifier introduction when one pair of variables is distinct. (Contributed by NM, 2-Jan-2002.)

Theoremdveel2 1973* Quantifier introduction when one pair of variables is distinct. (Contributed by NM, 2-Jan-2002.)

Theoremax15 1974 Axiom ax-15 2095 is redundant if we assume ax-17 1606. Remark 9.6 in [Megill] p. 448 (p. 16 of the preprint), regarding axiom scheme C14'.

Note that is a dummy variable introduced in the proof. On the web page, it is implicitly assumed to be distinct from all other variables. (This is made explicit in the database file set.mm). Its purpose is to satisfy the distinct variable requirements of dveel2 1973 and ax-17 1606. By the end of the proof it has vanished, and the final theorem has no distinct variable requirements. (Contributed by NM, 29-Jun-1995.) (Proof modification is discouraged.)

Theoremdrsb1 1975 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.)

Theoremsb2 1976 One direction of a simplified definition of substitution. (Contributed by NM, 5-Aug-1993.)

Theoremstdpc4 1977 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 spsbc 3016 and rspsbc 3082. (Contributed by NM, 5-Aug-1993.)

Theoremsbft 1978 Substitution has no effect on a non-free variable. (Contributed by NM, 30-May-2009.) (Revised by Mario Carneiro, 12-Oct-2016.)

Theoremsbf 1979 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 1980 Substitution for a variable not free in a wff does not affect it. (Contributed by NM, 5-Aug-1993.)

Theoremsbf2 1981 Substitution has no effect on a bound variable. (Contributed by NM, 1-Jul-2005.)

Theoremsb6x 1982 Equivalence involving substitution for a variable not free. (Contributed by NM, 5-Aug-1993.) (Revised by Mario Carneiro, 4-Oct-2016.)

Theoremnfs1f 1983 If is not free in , it is not free in . (Contributed by Mario Carneiro, 11-Aug-2016.)

Theoremsbequ5 1984 Substitution does not change an identical variable specifier. (Contributed by NM, 5-Aug-1993.)

Theoremsbequ6 1985 Substitution does not change a distinctor. (Contributed by NM, 5-Aug-1993.)

Theoremsbt 1986 A substitution into a theorem remains true. (See chvar 1939 and chvarv 1966 for versions using implicit substitution.) (Contributed by NM, 21-Jan-2004.) (Proof shortened by Andrew Salmon, 25-May-2011.)

Theoremequsb1 1987 Substitution applied to an atomic wff. (Contributed by NM, 5-Aug-1993.)

Theoremequsb2 1988 Substitution applied to an atomic wff. (Contributed by NM, 5-Aug-1993.)

Theoremsbied 1989 Conversion of implicit substitution to explicit substitution (deduction version of sbie 1991). (Contributed by NM, 30-Jun-1994.) (Revised by Mario Carneiro, 4-Oct-2016.)

Theoremsbiedv 1990* Conversion of implicit substitution to explicit substitution (deduction version of sbie 1991). (Contributed by NM, 7-Jan-2017.)

Theoremsbie 1991 Conversion of implicit substitution to explicit substitution. (Contributed by NM, 30-Jun-1994.) (Revised by Mario Carneiro, 4-Oct-2016.)

Theoremsb6f 1992 Equivalence for substitution when is not free in . (Contributed by NM, 5-Aug-1993.) (Revised by Mario Carneiro, 4-Oct-2016.)

Theoremsb5f 1993 Equivalence for substitution when is not free in . (Contributed by NM, 5-Aug-1993.) (Revised by Mario Carneiro, 4-Oct-2016.)

Theoremhbsb2a 1994 Special case of a bound-variable hypothesis builder for substitution. (Contributed by NM, 2-Feb-2007.)

Theoremhbsb2e 1995 Special case of a bound-variable hypothesis builder for substitution. (Contributed by NM, 2-Feb-2007.)

Theoremhbsb3 1996 If is not free in , is not free in . (Contributed by NM, 5-Aug-1993.)

Theoremnfs1 1997 If is not free in , is not free in . (Contributed by Mario Carneiro, 11-Aug-2016.)

Theoremax16 1998* Proof of older axiom ax-16 2096. (Contributed by NM, 8-Nov-2006.) (Revised by NM, 22-Sep-2017.)

Theoremax16i 1999* Inference with ax16 1998 as its conclusion. (Contributed by NM, 20-May-2008.) (Proof modification is discouraged.)

Theoremax16ALT 2000* Alternate proof of ax16 1998. (Contributed by NM, 17-May-2008.) (Proof modification is discouraged.) (New usage is discouraged.)

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