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Theorem List for Metamath Proof Explorer - 2101-2200   *Has distinct variable group(s)
TypeLabelDescription
Statement

Theoremequsb1 2101 Substitution applied to an atomic wff. (Contributed by NM, 5-Aug-1993.)

Theoremequsb2 2102 Substitution applied to an atomic wff. (Contributed by NM, 5-Aug-1993.)

Theoremcleljust 2103* When the class variables in definition df-clel 2432 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 1726 with the class variables in wcel 1725. 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 2104* When the class variables in definition df-clel 2432 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 1726 with the class variables in wcel 1725. (Contributed by NM, 28-Jan-2004.) (Revised by Mario Carneiro, 21-Dec-2016.) (Proof modification is discouraged.)

Theoremdveel1 2105* Quantifier introduction when one pair of variables is distinct. (Contributed by NM, 2-Jan-2002.)

Theoremdveel2 2106* Quantifier introduction when one pair of variables is distinct. (Contributed by NM, 2-Jan-2002.)

Theoremax15 2107 Axiom ax-15 2220 is redundant if we assume ax-17 1626. 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 2106 and ax-17 1626. 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.)

Theoremdfsb2 2108 An alternate definition of proper substitution that, like df-sb 1659, mixes free and bound variables to avoid distinct variable requirements. (Contributed by NM, 17-Feb-2005.)

Theoremdfsb3 2109 An alternate definition of proper substitution df-sb 1659 that uses only primitive connectives (no defined terms) on the right-hand side. (Contributed by NM, 6-Mar-2007.)

Theoremsbequi 2110 An equality theorem for substitution. (Contributed by NM, 5-Aug-1993.) (Proof shortened by Wolf Lammen, 14-Jul-2018.)

Theoremsbequ 2111 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.)

Theoremdrsb1 2112 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.)

Theoremdrsb2 2113 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.)

TheoremsbequiOLD 2114 Obsolete proof of sbequi 2110 as of 2-May-2018. (Contributed by NM, 5-Aug-1993.) (Proof modification is discoraged.) (New usage is discouraged.)

Theoremsbft 2115 Substitution has no effect on a non-free variable. (Contributed by NM, 30-May-2009.) (Revised by Mario Carneiro, 12-Oct-2016.) (Proof shortened by Wolf Lammen, 3-May-2018.)

TheoremsbftOLD 2116 Obsolete proof of sbft 2115 as of 22-Apr-2018. (Contributed by NM, 30-May-2009.) (Revised by Mario Carneiro, 12-Oct-2016.) (Proof modification is discouraged.) (New usage is discouraged.)

Theoremsbf 2117 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 2118 Substitution for a variable not free in a wff does not affect it. (Contributed by NM, 5-Aug-1993.)

Theoremsbf2 2119 Substitution has no effect on a bound variable. (Contributed by NM, 1-Jul-2005.)

Theoremnfs1f 2120 If is not free in , it is not free in . (Contributed by Mario Carneiro, 11-Aug-2016.)

Theoremsb6x 2121 Equivalence involving substitution for a variable not free. (Contributed by NM, 5-Aug-1993.) (Revised by Mario Carneiro, 4-Oct-2016.)

Theoremsb6f 2122 Equivalence for substitution when is not free in . (Contributed by NM, 5-Aug-1993.) (Revised by Mario Carneiro, 4-Oct-2016.)

Theoremsb5f 2123 Equivalence for substitution when is not free in . (Contributed by NM, 5-Aug-1993.) (Revised by Mario Carneiro, 4-Oct-2016.)

Theoremsbequ5 2124 Substitution does not change an identical variable specifier. (Contributed by NM, 5-Aug-1993.)

Theoremsbequ6 2125 Substitution does not change a distinctor. (Contributed by NM, 5-Aug-1993.)

Theoremsbt 2126 A substitution into a theorem remains true. (See chvar 1968 and chvarv 1969 for versions using implicit substitution.) (Contributed by NM, 21-Jan-2004.) (Proof shortened by Andrew Salmon, 25-May-2011.)

Theoremnfsb4t 2127 A variable not free remains so after substitution with a distinct variable (closed form of nfsb4 2129). (Contributed by NM, 7-Apr-2004.) (Revised by Mario Carneiro, 4-Oct-2016.) (Proof shortened by Wolf Lammen, 11-May-2018.)

Theoremnfsb4tOLD 2128 Obsolete proof of nfsb4t 2127 as of 6-May-2018. (Contributed by NM, 7-Apr-2004.) (Revised by Mario Carneiro, 4-Oct-2016.) (Proof modification is discouraged.) (New usage is discouraged.)

Theoremnfsb4 2129 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.)

Theoremsbn 2130 Negation inside and outside of substitution are equivalent. (Contributed by NM, 5-Aug-1993.) (Proof shortened by Wolf Lammen, 30-Apr-2018.)

TheoremsbnOLD 2131 Obsolete proof of sbn 2130 as of 30-Apr-2018. (Contributed by NM, 5-Aug-1993.) (Proof modification is discouraged.) (New usage is discouraged.)

Theoremsbi1 2132 Removal of implication from substitution. (Contributed by NM, 5-Aug-1993.)

Theoremsbi2 2133 Introduction of implication into substitution. (Contributed by NM, 5-Aug-1993.)

Theoremspsbim 2134 Specialization of implication. (Contributed by NM, 5-Aug-1993.) (Proof shortened by Andrew Salmon, 25-May-2011.)

Theoremsbim 2135 Implication inside and outside of substitution are equivalent. (Contributed by NM, 5-Aug-1993.)

Theoremsbrim 2136 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.)

Theoremsblim 2137 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.)

Theoremsbor 2138 Logical OR inside and outside of substitution are equivalent. (Contributed by NM, 29-Sep-2002.)

Theoremsban 2139 Conjunction inside and outside of a substitution are equivalent. (Contributed by NM, 5-Aug-1993.)

Theoremsb3an 2140 Conjunction inside and outside of a substitution are equivalent. (Contributed by NM, 14-Dec-2006.)

Theoremsbbi 2141 Equivalence inside and outside of a substitution are equivalent. (Contributed by NM, 5-Aug-1993.)

Theoremspsbbi 2142 Specialization of biconditional. (Contributed by NM, 5-Aug-1993.)

Theoremsbbid 2143 Deduction substituting both sides of a biconditional. (Contributed by NM, 5-Aug-1993.)

Theoremsblbis 2144 Introduce left biconditional inside of a substitution. (Contributed by NM, 19-Aug-1993.)

Theoremsbrbis 2145 Introduce right biconditional inside of a substitution. (Contributed by NM, 18-Aug-1993.)

Theoremsbrbif 2146 Introduce right biconditional inside of a substitution. (Contributed by NM, 18-Aug-1993.) (Revised by Mario Carneiro, 4-Oct-2016.)

TheoremspsbeOLD 2147 Obsolete proof of spsbe 1663 as of 3-May-2018. (Contributed by NM, 5-Aug-1993.) (Proof modification is discouraged.) (New usage is discouraged.)

Theoremsbequ8 2148 Elimination of equality from antecedent after substitution. (Contributed by NM, 5-Aug-1993.)

Theoremsbie 2149 Conversion of implicit substitution to explicit substitution. (Contributed by NM, 30-Jun-1994.) (Revised by Mario Carneiro, 4-Oct-2016.) (Revised by Wolf Lammen, 14-Jul-2018.)

Theoremsbied 2150 Conversion of implicit substitution to explicit substitution (deduction version of sbie 2149). (Contributed by NM, 30-Jun-1994.) (Revised by Mario Carneiro, 4-Oct-2016.) (Proof shortened by Wolf Lammen, 24-Jun-2018.)

TheoremsbiedOLD 2151 Obsolete proof of sbied 2150 as of 30-Apr-2018. (Contributed by NM, 30-Jun-1994.) (Revised by Mario Carneiro, 4-Oct-2016.) (Proof modification is discouraged.) (New usage is discouraged.)

TheoremsbieOLD 2152 Obsolete proof of sbie 2149 as of 30-Apr-2018. (Contributed by NM, 30-Jun-1994.) (Revised by Mario Carneiro, 4-Oct-2016.) (Proof modification is discouraged.) (New usage is discouraged.)

Theoremsbiedv 2153* Conversion of implicit substitution to explicit substitution (deduction version of sbie 2149). (Contributed by NM, 7-Jan-2017.)

Theoremax16ALT 2154* Alternate proof of ax16 2050. (Contributed by NM, 17-May-2008.) (Proof modification is discouraged.) (New usage is discouraged.)

Theoremax16ALT2 2155* Alternate proof of ax16 2050. (Contributed by NM, 8-Nov-2006.) (Proof modification is discouraged.) (New usage is discouraged.)

Theorema16gALT 2156* A generalization of axiom ax-16 2221. Alternate proof of a16g 2048 that uses df-sb 1659. (Contributed by NM, 5-Aug-1993.) (Proof shortened by Andrew Salmon, 25-May-2011.) (Proof modification is discouraged.) (New usage is discouraged.)

TheoremdvelimdfOLD 2157 Obsolete proof of dvelimdf 2070 as of 6-May-2018. (Contributed by NM, 7-Apr-2004.) (Revised by Mario Carneiro, 6-Oct-2016.) (Proof modification is discouraged.) (New usage is discouraged.)

Theoremsbco 2158 A composition law for substitution. (Contributed by NM, 5-Aug-1993.)

Theoremsbid2 2159 An identity law for substitution. (Contributed by NM, 5-Aug-1993.) (Revised by Mario Carneiro, 6-Oct-2016.)

Theoremsbidm 2160 An idempotent law for substitution. (Contributed by NM, 30-Jun-1994.) (Proof shortened by Andrew Salmon, 25-May-2011.)

Theoremsbco2 2161 A composition law for substitution. (Contributed by NM, 30-Jun-1994.) (Revised by Mario Carneiro, 6-Oct-2016.)

Theoremsbco2d 2162 A composition law for substitution. (Contributed by NM, 5-Aug-1993.) (Revised by Mario Carneiro, 6-Oct-2016.)

Theoremsbco3 2163 A composition law for substitution. (Contributed by NM, 5-Aug-1993.)

Theoremsbcom 2164 A commutativity law for substitution. (Contributed by NM, 27-May-1997.) (Proof shortened by Wolf Lammen, 24-Jun-2018.)

TheoremsbcomOLD 2165 Obsolete proof of sbcom 2164 as of 24-Jun-2018. (Contributed by NM, 27-May-1997.) (Proof modification is discouraged.) (New usage is discouraged.)

Theoremsb5rf 2166 Reversed substitution. (Contributed by NM, 3-Feb-2005.) (Revised by Mario Carneiro, 6-Oct-2016.)

Theoremsb6rf 2167 Reversed substitution. (Contributed by NM, 5-Aug-1993.) (Revised by Mario Carneiro, 6-Oct-2016.)

Theoremsb8 2168 Substitution of variable in universal quantifier. (Contributed by NM, 5-Aug-1993.) (Revised by Mario Carneiro, 6-Oct-2016.) (Proof shortened by Jim Kingdon, 15-Jan-2018.)

Theoremsb8e 2169 Substitution of variable in existential quantifier. (Contributed by NM, 12-Aug-1993.) (Revised by Mario Carneiro, 6-Oct-2016.) (Proof shortened by Jim Kingdon, 15-Jan-2018.)

Theoremsb9i 2170 Commutation of quantification and substitution variables. (Contributed by NM, 5-Aug-1993.)

Theoremsb9 2171 Commutation of quantification and substitution variables. (Contributed by NM, 5-Aug-1993.)

Theoremax11v 2172* This is a version of ax-11o 2218 when the variables are distinct. Axiom (C8) of [Monk2] p. 105. See theorem ax11v2 2078 for the rederivation of ax-11o 2218 from this theorem. (Contributed by NM, 5-Aug-1993.)

Theoremax11vALT 2173* Alternate proof of ax11v 2172 that avoids theorem ax16 2050 and is proved directly from ax-11 1761 rather than via ax11o 2081. (Contributed by Jim Kingdon, 15-Dec-2017.) (New usage is discouraged.) (Proof modification is discouraged.)

Theoremsb56 2174* 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 1659. (Contributed by NM, 14-Apr-2008.)

Theoremsb6 2175* 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.)

Theoremsb5 2176* Equivalence for substitution. Similar to Theorem 6.1 of [Quine] p. 40. (Contributed by NM, 18-Aug-1993.)

Theoremequsb3lem 2177* Lemma for equsb3 2178. (Contributed by Raph Levien and FL, 4-Dec-2005.) (Proof shortened by Andrew Salmon, 14-Jun-2011.)

Theoremequsb3 2178* Substitution applied to an atomic wff. (Contributed by Raph Levien and FL, 4-Dec-2005.)

Theoremelsb3 2179* Substitution applied to an atomic membership wff. (Contributed by NM, 7-Nov-2006.) (Proof shortened by Andrew Salmon, 14-Jun-2011.)

Theoremelsb4 2180* Substitution applied to an atomic membership wff. (Contributed by Rodolfo Medina, 3-Apr-2010.) (Proof shortened by Andrew Salmon, 14-Jun-2011.)

Theoremhbs1 2181* is not free in when and are distinct. (Contributed by NM, 5-Aug-1993.)

Theoremnfs1v 2182* is not free in when and are distinct. (Contributed by Mario Carneiro, 11-Aug-2016.)

Theoremsbhb 2183* Two ways of expressing " is (effectively) not free in ." (Contributed by NM, 29-May-2009.)

Theoremsbnf2 2184* Two ways of expressing " is (effectively) not free in ." (Contributed by Gérard Lang, 14-Nov-2013.) (Revised by Mario Carneiro, 6-Oct-2016.)

Theoremnfsb 2185* If is not free in , it is not free in when and are distinct. (Contributed by Mario Carneiro, 11-Aug-2016.)

Theoremhbsb 2186* If is not free in , it is not free in when and are distinct. (Contributed by NM, 12-Aug-1993.)

Theoremnfsbd 2187* Deduction version of nfsb 2185. (Contributed by NM, 15-Feb-2013.)

Theorem2sb5 2188* Equivalence for double substitution. (Contributed by NM, 3-Feb-2005.)

Theorem2sb6 2189* Equivalence for double substitution. (Contributed by NM, 3-Feb-2005.)

Theoremsbcom2 2190* Commutativity law for substitution. Used in proof of Theorem 9.7 of [Megill] p. 449 (p. 16 of the preprint). (Contributed by NM, 27-May-1997.)

Theorempm11.07 2191* (Probably not) Axiom *11.07 in [WhiteheadRussell] p. 159. The original confusingly reads: *11.07 "Whatever possible argument may be, is true whatever possible argument may be" implies the corresponding statement with and interchanged except in " ". This theorem will be deleted after 22-Feb-2018 if no one is able to determine the correct interpretation. See https://groups.google.com/d/msg/metamath/iS0fOvSemC8/YzrRyX70AgAJ. (Contributed by Andrew Salmon, 17-Jun-2011.) (Proof shortened by Jim Kingdon, 22-Jan-2018.) (New usage is discouraged.)

Theorempm11.07OLD 2192* Obsolete proof of pm11.07 2191 as of 22-Jan-2018. (Contributed by Andrew Salmon, 17-Jun-2011.) (New usage is discouraged.) (Proof modification is discouraged.)

Theoremsb6a 2193* Equivalence for substitution. (Contributed by NM, 5-Aug-1993.)

Theorem2sb5rf 2194* Reversed double substitution. (Contributed by NM, 3-Feb-2005.) (Revised by Mario Carneiro, 6-Oct-2016.)

Theorem2sb6rf 2195* Reversed double substitution. (Contributed by NM, 3-Feb-2005.) (Revised by Mario Carneiro, 6-Oct-2016.)

Theoremsb7f 2196* This version of dfsb7 2198 does not require that and be distinct. This permits it to be used as a definition for substitution in a formalization that omits the logically redundant axiom ax-17 1626 i.e. that doesn't have the concept of a variable not occurring in a wff. (df-sb 1659 is also suitable, but its mixing of free and bound variables is distasteful to some logicians.) (Contributed by NM, 26-Jul-2006.) (Revised by Mario Carneiro, 6-Oct-2016.)

Theoremsb7h 2197* This version of dfsb7 2198 does not require that and be distinct. This permits it to be used as a definition for substitution in a formalization that omits the logically redundant axiom ax-17 1626 i.e. that doesn't have the concept of a variable not occurring in a wff. (df-sb 1659 is also suitable, but its mixing of free and bound variables is distasteful to some logicians.) (Contributed by NM, 26-Jul-2006.) (Proof shortened by Andrew Salmon, 25-May-2011.)

Theoremdfsb7 2198* An alternate definition of proper substitution df-sb 1659. By introducing a dummy variable in the definiens, we are able to eliminate any distinct variable restrictions among the variables , , and of the definiendum. No distinct variable conflicts arise because effectively insulates from . To achieve this, we use a chain of two substitutions in the form of sb5 2176, first for then for . Compare Definition 2.1'' of [Quine] p. 17, which is obtained from this theorem by applying df-clab 2423. Theorem sb7h 2197 provides a version where and don't have to be distinct. (Contributed by NM, 28-Jan-2004.)

Theoremsb10f 2199* Hao Wang's identity axiom P6 in Irving Copi, Symbolic Logic (5th ed., 1979), p. 328. In traditional predicate calculus, this is a sole axiom for identity from which the usual ones can be derived. (Contributed by NM, 9-May-2005.) (Revised by Mario Carneiro, 6-Oct-2016.)

Theoremsbid2v 2200* An identity law for substitution. Used in proof of Theorem 9.7 of [Megill] p. 449 (p. 16 of the preprint). (Contributed by NM, 5-Aug-1993.)

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