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Type | Label | Description |
---|---|---|
Statement | ||
Theorem | equvin 2001* | A variable introduction law for equality. Lemma 15 of [Monk2] p. 109. (Contributed by NM, 5-Aug-1993.) |
Theorem | cbvalv 2002* | Rule used to change bound variables, using implicit substitution. (Contributed by NM, 5-Aug-1993.) |
Theorem | cbvexv 2003* | Rule used to change bound variables, using implicit substitution. (Contributed by NM, 5-Aug-1993.) |
Theorem | cbval2 2004* | Rule used to change bound variables, using implicit substitution. (Contributed by NM, 22-Dec-2003.) (Revised by Mario Carneiro, 6-Oct-2016.) |
Theorem | cbvex2 2005* | Rule used to change bound variables, using implicit substitution. (Contributed by NM, 14-Sep-2003.) (Revised by Mario Carneiro, 6-Oct-2016.) |
Theorem | cbval2v 2006* | Rule used to change bound variables, using implicit substitution. (Contributed by NM, 4-Feb-2005.) |
Theorem | cbvex2v 2007* | Rule used to change bound variables, using implicit substitution. (Contributed by NM, 26-Jul-1995.) |
Theorem | cbvald 2008* | Deduction used to change bound variables, using implicit substitution, particularly useful in conjunction with dvelim 2016. (Contributed by NM, 2-Jan-2002.) (Revised by Mario Carneiro, 6-Oct-2016.) |
Theorem | cbvexd 2009* | Deduction used to change bound variables, using implicit substitution, particularly useful in conjunction with dvelim 2016. (Contributed by NM, 2-Jan-2002.) (Revised by Mario Carneiro, 6-Oct-2016.) |
Theorem | cbvaldva 2010* | Rule used to change the bound variable in a universal quantifier with implicit substitution. Deduction form. (Contributed by David Moews, 1-May-2017.) |
Theorem | cbvexdva 2011* | Rule used to change the bound variable in an existential quantifier with implicit substitution. Deduction form. (Contributed by David Moews, 1-May-2017.) |
Theorem | cbvex4v 2012* | Rule used to change bound variables, using implicit substitution. (Contributed by NM, 26-Jul-1995.) |
Theorem | chvarv 2013* | Implicit substitution of for into a theorem. (Contributed by NM, 20-Apr-1994.) |
Theorem | cleljust 2014* | When the class variables in definition df-clel 2349 are replaced with setvar 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 setvar variables in wel 1711 with the class variables in wcel 1710. 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.) |
Theorem | cleljustALT 2015* | When the class variables in definition df-clel 2349 are replaced with setvar 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 setvar variables in wel 1711 with the class variables in wcel 1710. (Contributed by NM, 28-Jan-2004.) (Proof modification is discouraged.) (New usage is discouraged.) (Revised by Mario Carneiro, 21-Dec-2016.) |
Theorem | dvelim 2016* |
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 2082. Other variants of this theorem are dvelimh 1964 (with no distinct variable restrictions), dvelimhw 1849 (that avoids ax-12 1925), and dvelimALT 2133 (that avoids ax-10 2140). (Contributed by NM, 23-Nov-1994.) |
Theorem | dvelimnf 2017* | Version of dvelim 2016 using "not free" notation. (Contributed by Mario Carneiro, 9-Oct-2016.) |
Theorem | dveeq1 2018* | Quantifier introduction when one pair of variables is distinct. (Contributed by NM, 2-Jan-2002.) |
Theorem | dveel1 2019* | Quantifier introduction when one pair of variables is distinct. (Contributed by NM, 2-Jan-2002.) |
Theorem | dveel2 2020* | Quantifier introduction when one pair of variables is distinct. (Contributed by NM, 2-Jan-2002.) |
Theorem | ax15 2021 |
Axiom ax-15 2143 is redundant if we assume ax-17 1616. 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 2020 and ax-17 1616. 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.) |
Theorem | drsb1 2022 | 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.) |
Theorem | sb2 2023 | One direction of a simplified definition of substitution. (Contributed by NM, 5-Aug-1993.) |
Theorem | stdpc4 2024 | 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 3058 and rspsbc 3124. (Contributed by NM, 5-Aug-1993.) |
Theorem | sbft 2025 | Substitution has no effect on a non-free variable. (Contributed by NM, 30-May-2009.) (Revised by Mario Carneiro, 12-Oct-2016.) |
Theorem | sbf 2026 | 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.) |
Theorem | sbh 2027 | Substitution for a variable not free in a wff does not affect it. (Contributed by NM, 5-Aug-1993.) |
Theorem | sbf2 2028 | Substitution has no effect on a bound variable. (Contributed by NM, 1-Jul-2005.) |
Theorem | sb6x 2029 | Equivalence involving substitution for a variable not free. (Contributed by NM, 5-Aug-1993.) (Revised by Mario Carneiro, 4-Oct-2016.) |
Theorem | nfs1f 2030 | If is not free in , it is not free in . (Contributed by Mario Carneiro, 11-Aug-2016.) |
Theorem | sbequ5 2031 | Substitution does not change an identical variable specifier. (Contributed by NM, 5-Aug-1993.) |
Theorem | sbequ6 2032 | Substitution does not change a distinctor. (Contributed by NM, 5-Aug-1993.) |
Theorem | sbt 2033 | A substitution into a theorem remains true. (See chvar 1986 and chvarv 2013 for versions using implicit substitution.) (Contributed by NM, 21-Jan-2004.) (Proof shortened by Andrew Salmon, 25-May-2011.) |
Theorem | equsb1 2034 | Substitution applied to an atomic wff. (Contributed by NM, 5-Aug-1993.) |
Theorem | equsb2 2035 | Substitution applied to an atomic wff. (Contributed by NM, 5-Aug-1993.) |
Theorem | sbied 2036 | Conversion of implicit substitution to explicit substitution (deduction version of sbie 2038). (Contributed by NM, 30-Jun-1994.) (Revised by Mario Carneiro, 4-Oct-2016.) |
Theorem | sbiedv 2037* | Conversion of implicit substitution to explicit substitution (deduction version of sbie 2038). (Contributed by NM, 7-Jan-2017.) |
Theorem | sbie 2038 | Conversion of implicit substitution to explicit substitution. (Contributed by NM, 30-Jun-1994.) (Revised by Mario Carneiro, 4-Oct-2016.) |
Theorem | sb6f 2039 | Equivalence for substitution when is not free in . (Contributed by NM, 5-Aug-1993.) (Revised by Mario Carneiro, 4-Oct-2016.) |
Theorem | sb5f 2040 | Equivalence for substitution when is not free in . (Contributed by NM, 5-Aug-1993.) (Revised by Mario Carneiro, 4-Oct-2016.) |
Theorem | hbsb2a 2041 | Special case of a bound-variable hypothesis builder for substitution. (Contributed by NM, 2-Feb-2007.) |
Theorem | hbsb2e 2042 | Special case of a bound-variable hypothesis builder for substitution. (Contributed by NM, 2-Feb-2007.) |
Theorem | hbsb3 2043 | If is not free in , is not free in . (Contributed by NM, 5-Aug-1993.) |
Theorem | nfs1 2044 | If is not free in , is not free in . (Contributed by Mario Carneiro, 11-Aug-2016.) |
Theorem | ax16 2045* | Proof of older axiom ax-16 2144. (Contributed by NM, 8-Nov-2006.) (Revised by NM, 22-Sep-2017.) |
Theorem | ax16i 2046* | Inference with ax16 2045 as its conclusion. (Contributed by NM, 20-May-2008.) (Proof modification is discouraged.) |
Theorem | ax16ALT 2047* | Alternate proof of ax16 2045. (Contributed by NM, 17-May-2008.) (Proof modification is discouraged.) (New usage is discouraged.) |
Theorem | ax16ALT2 2048* | Alternate proof of ax16 2045. (Contributed by NM, 8-Nov-2006.) (Proof modification is discouraged.) (New usage is discouraged.) |
Theorem | a16gALT 2049* | A generalization of axiom ax-16 2144. Alternate proof of a16g 1945 that uses df-sb 1649. (Contributed by NM, 5-Aug-1993.) (Proof shortened by Andrew Salmon, 25-May-2011.) (Proof modification is discouraged.) (New usage is discouraged.) |
Theorem | a16gb 2050* | A generalization of axiom ax-16 2144. (Contributed by NM, 5-Aug-1993.) |
Theorem | a16nf 2051* | If dtru in set.mm is false, then there is only one element in the universe, so everything satisfies . (Contributed by Mario Carneiro, 7-Oct-2016.) |
Theorem | sb3 2052 | One direction of a simplified definition of substitution when variables are distinct. (Contributed by NM, 5-Aug-1993.) |
Theorem | sb4 2053 | One direction of a simplified definition of substitution when variables are distinct. (Contributed by NM, 5-Aug-1993.) |
Theorem | sb4b 2054 | Simplified definition of substitution when variables are distinct. (Contributed by NM, 27-May-1997.) |
Theorem | dfsb2 2055 | An alternate definition of proper substitution that, like df-sb 1649, mixes free and bound variables to avoid distinct variable requirements. (Contributed by NM, 17-Feb-2005.) |
Theorem | dfsb3 2056 | An alternate definition of proper substitution df-sb 1649 that uses only primitive connectives (no defined terms) on the right-hand side. (Contributed by NM, 6-Mar-2007.) |
Theorem | hbsb2 2057 | Bound-variable hypothesis builder for substitution. (Contributed by NM, 5-Aug-1993.) |
Theorem | nfsb2 2058 | Bound-variable hypothesis builder for substitution. (Contributed by Mario Carneiro, 4-Oct-2016.) |
Theorem | sbequi 2059 | An equality theorem for substitution. (Contributed by NM, 5-Aug-1993.) |
Theorem | sbequ 2060 | An equality theorem for substitution. Used in proof of Theorem 9.7 in [Megill] p. 449 (p. 16 of the preprint). (Contributed by NM, 5-Aug-1993.) |
Theorem | drsb2 2061 | 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.) |
Theorem | sbn 2062 | Negation inside and outside of substitution are equivalent. (Contributed by NM, 5-Aug-1993.) |
Theorem | sbi1 2063 | Removal of implication from substitution. (Contributed by NM, 5-Aug-1993.) |
Theorem | sbi2 2064 | Introduction of implication into substitution. (Contributed by NM, 5-Aug-1993.) |
Theorem | sbim 2065 | Implication inside and outside of substitution are equivalent. (Contributed by NM, 5-Aug-1993.) |
Theorem | sbor 2066 | Logical OR inside and outside of substitution are equivalent. (Contributed by NM, 29-Sep-2002.) |
Theorem | sbrim 2067 | Substitution with a variable not free in antecedent affects only the consequent. (Contributed by NM, 5-Aug-1993.) (Revised by Mario Carneiro, 4-Oct-2016.) |
Theorem | sblim 2068 | Substitution with a variable not free in consequent affects only the antecedent. (Contributed by NM, 14-Nov-2013.) (Revised by Mario Carneiro, 4-Oct-2016.) |
Theorem | sban 2069 | Conjunction inside and outside of a substitution are equivalent. (Contributed by NM, 5-Aug-1993.) |
Theorem | sb3an 2070 | Conjunction inside and outside of a substitution are equivalent. (Contributed by NM, 14-Dec-2006.) |
Theorem | sbbi 2071 | Equivalence inside and outside of a substitution are equivalent. (Contributed by NM, 5-Aug-1993.) |
Theorem | sblbis 2072 | Introduce left biconditional inside of a substitution. (Contributed by NM, 19-Aug-1993.) |
Theorem | sbrbis 2073 | Introduce right biconditional inside of a substitution. (Contributed by NM, 18-Aug-1993.) |
Theorem | sbrbif 2074 | Introduce right biconditional inside of a substitution. (Contributed by NM, 18-Aug-1993.) (Revised by Mario Carneiro, 4-Oct-2016.) |
Theorem | spsbe 2075 | A specialization theorem. (Contributed by NM, 5-Aug-1993.) |
Theorem | spsbim 2076 | Specialization of implication. (Contributed by NM, 5-Aug-1993.) (Proof shortened by Andrew Salmon, 25-May-2011.) |
Theorem | spsbbi 2077 | Specialization of biconditional. (Contributed by NM, 5-Aug-1993.) |
Theorem | sbbid 2078 | Deduction substituting both sides of a biconditional. (Contributed by NM, 5-Aug-1993.) |
Theorem | sbequ8 2079 | Elimination of equality from antecedent after substitution. (Contributed by NM, 5-Aug-1993.) |
Theorem | nfsb4t 2080 | A variable not free remains so after substitution with a distinct variable (closed form of nfsb4 2081). (Contributed by NM, 7-Apr-2004.) (Revised by Mario Carneiro, 4-Oct-2016.) |
Theorem | nfsb4 2081 | A variable not free remains so after substitution with a distinct variable. (Contributed by NM, 5-Aug-1993.) (Revised by Mario Carneiro, 4-Oct-2016.) |
Theorem | dvelimdf 2082 | Deduction form of dvelimf 1997. This version may be useful if we want to avoid ax-17 1616 and use ax-16 2144 instead. (Contributed by NM, 7-Apr-2004.) (Revised by Mario Carneiro, 6-Oct-2016.) |
Theorem | sbco 2083 | A composition law for substitution. (Contributed by NM, 5-Aug-1993.) |
Theorem | sbid2 2084 | An identity law for substitution. (Contributed by NM, 5-Aug-1993.) (Revised by Mario Carneiro, 6-Oct-2016.) |
Theorem | sbidm 2085 | An idempotent law for substitution. (Contributed by NM, 30-Jun-1994.) (Proof shortened by Andrew Salmon, 25-May-2011.) |
Theorem | sbco2 2086 | A composition law for substitution. (Contributed by NM, 30-Jun-1994.) (Revised by Mario Carneiro, 6-Oct-2016.) |
Theorem | sbco2d 2087 | A composition law for substitution. (Contributed by NM, 5-Aug-1993.) (Revised by Mario Carneiro, 6-Oct-2016.) |
Theorem | sbco3 2088 | A composition law for substitution. (Contributed by NM, 5-Aug-1993.) |
Theorem | sbcom 2089 | A commutativity law for substitution. (Contributed by NM, 27-May-1997.) |
Theorem | sb5rf 2090 | Reversed substitution. (Contributed by NM, 3-Feb-2005.) (Revised by Mario Carneiro, 6-Oct-2016.) |
Theorem | sb6rf 2091 | Reversed substitution. (Contributed by NM, 5-Aug-1993.) (Revised by Mario Carneiro, 6-Oct-2016.) |
Theorem | sb8 2092 | Substitution of variable in universal quantifier. (Contributed by NM, 5-Aug-1993.) (Revised by Mario Carneiro, 6-Oct-2016.) |
Theorem | sb8e 2093 | Substitution of variable in existential quantifier. (Contributed by NM, 12-Aug-1993.) (Revised by Mario Carneiro, 6-Oct-2016.) |
Theorem | sb9i 2094 | Commutation of quantification and substitution variables. (Contributed by NM, 5-Aug-1993.) |
Theorem | sb9 2095 | Commutation of quantification and substitution variables. (Contributed by NM, 5-Aug-1993.) |
Theorem | ax11v 2096* | This is a version of ax-11o 2141 when the variables are distinct. Axiom (C8) of [Monk2] p. 105. See theorem ax11v2 1992 for the rederivation of ax-11o 2141 from this theorem. (Contributed by NM, 5-Aug-1993.) |
Theorem | ax11vALT 2097* | Alternate proof of ax11v 2096 that avoids theorem ax16 2045 and is proved directly from ax-11 1746 rather than via ax11o 1994. (Contributed by Jim Kingdon, 15-Dec-2017.) (New usage is discouraged.) (Proof modification is discouraged.) |
Theorem | sb56 2098* | Two equivalent ways of expressing the proper substitution of for in , when and are distinct. Theorem 6.2 of [Quine] p. 40. The proof does not involve df-sb 1649. (Contributed by NM, 14-Apr-2008.) |
Theorem | sb6 2099* | Equivalence for substitution. Compare Theorem 6.2 of [Quine] p. 40. Also proved as Lemmas 16 and 17 of [Tarski] p. 70. (Contributed by NM, 18-Aug-1993.) |
Theorem | sb5 2100* | Equivalence for substitution. Similar to Theorem 6.1 of [Quine] p. 40. (Contributed by NM, 18-Aug-1993.) |
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