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

Theoremhbsb2a 2101 Special case of a bound-variable hypothesis builder for substitution. (Contributed by NM, 2-Feb-2007.)

Theoremhbsb2e 2102 Special case of a bound-variable hypothesis builder for substitution. (Contributed by NM, 2-Feb-2007.)

Theoremhbsb3 2103 If is not free in , is not free in . (Contributed by NM, 5-Aug-1993.)

Theoremnfs1 2104 If is not free in , is not free in . (Contributed by Mario Carneiro, 11-Aug-2016.)

Theoremequsb1 2105 Substitution applied to an atomic wff. (Contributed by NM, 5-Aug-1993.)

Theoremequsb2 2106 Substitution applied to an atomic wff. (Contributed by NM, 5-Aug-1993.)

Theoremcleljust 2107* When the class variables in definition df-clel 2438 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 1728 with the class variables in wcel 1727. 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 2108* When the class variables in definition df-clel 2438 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 1728 with the class variables in wcel 1727. (Contributed by NM, 28-Jan-2004.) (Revised by Mario Carneiro, 21-Dec-2016.) (Proof modification is discouraged.)

Theoremdveel1 2109* Quantifier introduction when one pair of variables is distinct. (Contributed by NM, 2-Jan-2002.)

Theoremdveel2 2110* Quantifier introduction when one pair of variables is distinct. (Contributed by NM, 2-Jan-2002.)

Theoremax15 2111 Axiom ax-15 2226 is redundant if we assume ax-17 1627. 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 2110 and ax-17 1627. 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 2112 An alternate definition of proper substitution that, like df-sb 1660, mixes free and bound variables to avoid distinct variable requirements. (Contributed by NM, 17-Feb-2005.)

Theoremdfsb3 2113 An alternate definition of proper substitution df-sb 1660 that uses only primitive connectives (no defined terms) on the right-hand side. (Contributed by NM, 6-Mar-2007.)

Theoremsbequi 2114 An equality theorem for substitution. (Contributed by NM, 5-Aug-1993.) (Proof shortened by Wolf Lammen, 14-Jul-2018.)

Theoremsbequ 2115 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 2116 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 2117 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 2118 Obsolete proof of sbequi 2114 as of 2-May-2018. (Contributed by NM, 5-Aug-1993.) (Proof modification is discoraged.) (New usage is discouraged.)

Theoremsbft 2119 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 2120 Obsolete proof of sbft 2119 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 2121 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 2122 Substitution for a variable not free in a wff does not affect it. (Contributed by NM, 5-Aug-1993.)

Theoremsbf2 2123 Substitution has no effect on a bound variable. (Contributed by NM, 1-Jul-2005.)

Theoremnfs1f 2124 If is not free in , it is not free in . (Contributed by Mario Carneiro, 11-Aug-2016.)

Theoremsb6x 2125 Equivalence involving substitution for a variable not free. (Contributed by NM, 5-Aug-1993.) (Revised by Mario Carneiro, 4-Oct-2016.)

Theoremsb6f 2126 Equivalence for substitution when is not free in . (Contributed by NM, 5-Aug-1993.) (Revised by Mario Carneiro, 4-Oct-2016.)

Theoremsb5f 2127 Equivalence for substitution when is not free in . (Contributed by NM, 5-Aug-1993.) (Revised by Mario Carneiro, 4-Oct-2016.)

Theoremsbequ5 2128 Substitution does not change an identical variable specifier. (Contributed by NM, 5-Aug-1993.)

Theoremsbequ6 2129 Substitution does not change a distinctor. (Contributed by NM, 5-Aug-1993.)

TheoremsbtOLD 2130 Obsolete proof of sbt 2095 as of 20-Jul-2018.) (Contributed by NM, 21-Jan-2004.) (Proof shortened by Andrew Salmon, 25-May-2011.) (Proof modification is discouraged.) (New usage is discouraged.)

Theoremnfsb4t 2131 A variable not free remains so after substitution with a distinct variable (closed form of nfsb4 2133). (Contributed by NM, 7-Apr-2004.) (Revised by Mario Carneiro, 4-Oct-2016.) (Proof shortened by Wolf Lammen, 11-May-2018.)

Theoremnfsb4tOLD 2132 Obsolete proof of nfsb4t 2131 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 2133 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 2134 Negation inside and outside of substitution are equivalent. (Contributed by NM, 5-Aug-1993.) (Proof shortened by Wolf Lammen, 30-Apr-2018.)

TheoremsbnOLD 2135 Obsolete proof of sbn 2134 as of 30-Apr-2018. (Contributed by NM, 5-Aug-1993.) (Proof modification is discouraged.) (New usage is discouraged.)

Theoremsbi1 2136 Removal of implication from substitution. (Contributed by NM, 5-Aug-1993.)

Theoremsbi2 2137 Introduction of implication into substitution. (Contributed by NM, 5-Aug-1993.)

Theoremspsbim 2138 Specialization of implication. (Contributed by NM, 5-Aug-1993.) (Proof shortened by Andrew Salmon, 25-May-2011.)

Theoremsbim 2139 Implication inside and outside of substitution are equivalent. (Contributed by NM, 5-Aug-1993.)

Theoremsbrim 2140 Substitution with a variable not free in antecedent affects only the consequent. Revised to minimize dependencies. (Contributed by NM, 5-Aug-1993.) (Revised by Mario Carneiro, 4-Oct-2016.) (Revised by Wolf Lammen, 28-Jul-2018.)

TheoremsbrimALT 2141 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.) (New usage is discouraged.)

Theoremsblim 2142 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 2143 Logical OR inside and outside of substitution are equivalent. (Contributed by NM, 29-Sep-2002.)

Theoremsban 2144 Conjunction inside and outside of a substitution are equivalent. (Contributed by NM, 5-Aug-1993.)

Theoremsb3an 2145 Conjunction inside and outside of a substitution are equivalent. (Contributed by NM, 14-Dec-2006.)

Theoremsbbi 2146 Equivalence inside and outside of a substitution are equivalent. (Contributed by NM, 5-Aug-1993.)

Theoremspsbbi 2147 Specialization of biconditional. (Contributed by NM, 5-Aug-1993.)

Theoremsbbid 2148 Deduction substituting both sides of a biconditional. (Contributed by NM, 5-Aug-1993.)

Theoremsblbis 2149 Introduce left biconditional inside of a substitution. (Contributed by NM, 19-Aug-1993.)

Theoremsbrbis 2150 Introduce right biconditional inside of a substitution. (Contributed by NM, 18-Aug-1993.)

Theoremsbrbif 2151 Introduce right biconditional inside of a substitution. (Contributed by NM, 18-Aug-1993.) (Revised by Mario Carneiro, 4-Oct-2016.)

TheoremspsbeOLD 2152 Obsolete proof of spsbe 1664 as of 3-May-2018. (Contributed by NM, 5-Aug-1993.) (Proof modification is discouraged.) (New usage is discouraged.)

Theoremsbequ8ALT 2153 Elimination of equality from antecedent after substitution. (Contributed by NM, 5-Aug-1993.) (New usage is discouraged.)

Theoremsbie 2154 Conversion of implicit substitution to explicit substitution. (Contributed by NM, 30-Jun-1994.) (Revised by Mario Carneiro, 4-Oct-2016.) (Revised by Wolf Lammen, 30-Apr-2018.)

TheoremsbieALT 2155 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 2156 Conversion of implicit substitution to explicit substitution (deduction version of sbie 2154). (Contributed by NM, 30-Jun-1994.) (Revised by Mario Carneiro, 4-Oct-2016.) (Proof shortened by Wolf Lammen, 24-Jun-2018.)

TheoremsbiedOLD 2157 Obsolete proof of sbied 2156 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 2158 Obsolete proof of sbie 2154 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 2159* Conversion of implicit substitution to explicit substitution (deduction version of sbie 2154). (Contributed by NM, 7-Jan-2017.)

Theoremax16ALT 2160* Alternate proof of ax16 2053. (Contributed by NM, 17-May-2008.) (Proof modification is discouraged.) (New usage is discouraged.)

Theoremax16ALT2 2161* Alternate proof of ax16 2053. (Contributed by NM, 8-Nov-2006.) (Proof modification is discouraged.) (New usage is discouraged.)

Theorema16gALT 2162* A generalization of axiom ax-16 2227. Alternate proof of a16g 2051 that uses df-sb 1660. (Contributed by NM, 5-Aug-1993.) (Proof shortened by Andrew Salmon, 25-May-2011.) (Proof modification is discouraged.) (New usage is discouraged.)

TheoremdvelimdfOLD 2163 Obsolete proof of dvelimdf 2073 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 2164 A composition law for substitution. (Contributed by NM, 5-Aug-1993.)

Theoremsbid2 2165 An identity law for substitution. (Contributed by NM, 5-Aug-1993.) (Revised by Mario Carneiro, 6-Oct-2016.)

Theoremsbidm 2166 An idempotent law for substitution. (Contributed by NM, 30-Jun-1994.) (Proof shortened by Andrew Salmon, 25-May-2011.)

Theoremsbco2 2167 A composition law for substitution. (Contributed by NM, 30-Jun-1994.) (Revised by Mario Carneiro, 6-Oct-2016.)

Theoremsbco2d 2168 A composition law for substitution. (Contributed by NM, 5-Aug-1993.) (Revised by Mario Carneiro, 6-Oct-2016.)

Theoremsbco3 2169 A composition law for substitution. (Contributed by NM, 5-Aug-1993.)

Theoremsbcom 2170 A commutativity law for substitution. (Contributed by NM, 27-May-1997.) (Proof shortened by Wolf Lammen, 24-Jun-2018.)

TheoremsbcomOLD 2171 Obsolete proof of sbcom 2170 as of 24-Jun-2018. (Contributed by NM, 27-May-1997.) (Proof modification is discouraged.) (New usage is discouraged.)

Theoremsb5rf 2172 Reversed substitution. (Contributed by NM, 3-Feb-2005.) (Revised by Mario Carneiro, 6-Oct-2016.)

Theoremsb6rf 2173 Reversed substitution. (Contributed by NM, 5-Aug-1993.) (Revised by Mario Carneiro, 6-Oct-2016.)

Theoremsb8 2174 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 2175 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 2176 Commutation of quantification and substitution variables. (Contributed by NM, 5-Aug-1993.)

Theoremsb9 2177 Commutation of quantification and substitution variables. (Contributed by NM, 5-Aug-1993.)

Theoremax11v 2178* This is a version of ax-11o 2224 when the variables are distinct. Axiom (C8) of [Monk2] p. 105. See theorem ax11v2 2081 for the rederivation of ax-11o 2224 from this theorem. (Contributed by NM, 5-Aug-1993.)

Theoremax11vALT 2179* Alternate proof of ax11v 2178 that avoids theorem ax16 2053 and is proved directly from ax-11 1763 rather than via ax11o 2084. (Contributed by Jim Kingdon, 15-Dec-2017.) (New usage is discouraged.) (Proof modification is discouraged.)

Theoremsb56 2180* 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 1660. (Contributed by NM, 14-Apr-2008.)

Theoremsb6 2181* 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 2182* Equivalence for substitution. Similar to Theorem 6.1 of [Quine] p. 40. (Contributed by NM, 18-Aug-1993.)

Theoremequsb3lem 2183* Lemma for equsb3 2184. (Contributed by Raph Levien and FL, 4-Dec-2005.) (Proof shortened by Andrew Salmon, 14-Jun-2011.)

Theoremequsb3 2184* Substitution applied to an atomic wff. (Contributed by Raph Levien and FL, 4-Dec-2005.)

Theoremelsb3 2185* Substitution applied to an atomic membership wff. (Contributed by NM, 7-Nov-2006.) (Proof shortened by Andrew Salmon, 14-Jun-2011.)

Theoremelsb4 2186* Substitution applied to an atomic membership wff. (Contributed by Rodolfo Medina, 3-Apr-2010.) (Proof shortened by Andrew Salmon, 14-Jun-2011.)

Theoremhbs1 2187* is not free in when and are distinct. (Contributed by NM, 5-Aug-1993.)

Theoremnfs1v 2188* is not free in when and are distinct. (Contributed by Mario Carneiro, 11-Aug-2016.)

Theoremsbhb 2189* Two ways of expressing " is (effectively) not free in ." (Contributed by NM, 29-May-2009.)

Theoremsbnf2 2190* Two ways of expressing " is (effectively) not free in ." (Contributed by Gérard Lang, 14-Nov-2013.) (Revised by Mario Carneiro, 6-Oct-2016.)

Theoremnfsb 2191* If is not free in , it is not free in when and are distinct. (Contributed by Mario Carneiro, 11-Aug-2016.)

Theoremhbsb 2192* If is not free in , it is not free in when and are distinct. (Contributed by NM, 12-Aug-1993.)

Theoremnfsbd 2193* Deduction version of nfsb 2191. (Contributed by NM, 15-Feb-2013.)

Theorem2sb5 2194* Equivalence for double substitution. (Contributed by NM, 3-Feb-2005.)

Theorem2sb6 2195* Equivalence for double substitution. (Contributed by NM, 3-Feb-2005.)

Theoremsbcom2 2196* 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 2197* (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 2198* Obsolete proof of pm11.07 2197 as of 22-Jan-2018. (Contributed by Andrew Salmon, 17-Jun-2011.) (New usage is discouraged.) (Proof modification is discouraged.)

Theoremsb6a 2199* Equivalence for substitution. (Contributed by NM, 5-Aug-1993.)

Theorem2sb5rf 2200* Reversed double substitution. (Contributed by NM, 3-Feb-2005.) (Revised by Mario Carneiro, 6-Oct-2016.)

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