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Theorem List for Metamath Proof Explorer - 2001-2100   *Has distinct variable group(s)
TypeLabelDescription
Statement

Theoremequsalh 2001 A useful equivalence related to substitution. (Contributed by NM, 5-Aug-1993.)

Theoremequsex 2002 A useful equivalence related to substitution. (Contributed by NM, 5-Aug-1993.) (Revised by Mario Carneiro, 3-Oct-2016.) (Proof shortened by Wolf Lammen, 6-Feb-2018.)

TheoremequsexOLD 2003 Obsolete proof of equsex 2002 as of 6-Feb-2018. (Contributed by NM, 5-Aug-1993.) (Revised by Mario Carneiro, 3-Oct-2016.) (New usage is discouraged.) (Proof modification is discouraged.)

Theoremequsexh 2004 A useful equivalence related to substitution. (Contributed by NM, 5-Aug-1993.)

Theoremax12olem1 2005* Lemma for nfeqf 2009 and dveeq1 2021. Used to eliminate distinct variable constraints. The proof of ax12o 2010 bases on ideas from NM, 24-Dec-2015. (Contributed by Wolf Lammen, 8-Feb-2018.)

Theoremax12olem2 2006* Lemma for nfeqf 2009 and dveeq1 2021. This lemma is equivalent to ax12v 1951 with one distinct variable constraint removed. (Contributed by Wolf Lammen, 29-Apr-2018.)

Theoremax12olem3 2007* Lemma for nfeqf 2009 and dveeq1 2021. Convert ax12olem2 2006 into a more general form. (Contributed by Wolf Lammen, 29-Apr-2018.)

Theoremax12olem4 2008* Lemma for nfeqf 2009. A technical step to remove a distinct variable constraint from ax12v 1951. (Contributed by Wolf Lammen, 29-Apr-2018.)

Theoremnfeqf 2009 A variable is effectively not free in an equality if it is not either of the involved variables. version of ax-12o 2219. (Contributed by Mario Carneiro, 6-Oct-2016.) (Proof shortened by Wolf Lammen, 29-Apr-2018.)

Theoremax12o 2010 Derive set.mm's original ax-12o 2219 from the shorter ax-12 1950. (Contributed by NM, 29-Nov-2015.) (Revised by NM, 24-Dec-2015.) (Proof shortened by Wolf Lammen, 29-Apr-2018.)

Theoremax12olem1OLD 2011* Obsolete proof of ax12oOLD 2018 as of 30-Jan-2018. Lemma for ax12oOLD 2018. Similar to equvin 2087 but with a negated equality. (Contributed by NM, 24-Dec-2015.) (Proof shortened by Wolf Lammen, 20-Jan-2018.) (New usage is discouraged.) (Proof modification is discouraged.)

Theoremax12olem2OLD 2012* Obsolete proof of ax12oOLD 2018 as of 30-Jan-2018. Lemma for ax12oOLD 2018. Negate the equalities in ax-12 1950, shown as the hypothesis. (Contributed by NM, 24-Dec-2015.) (Proof shortened by Wolf Lammen, 23-Jan-2018.) (New usage is discouraged.) (Proof modification is discouraged.)

Theoremax12olem3OLD 2013 Obsolete proof of ax12oOLD 2018 as of 30-Jan-2018. Lemma for ax12oOLD 2018. Show the equivalence of an intermediate equivalent to ax12o 2010 with the conjunction of ax-12 1950 and a variant with negated equalities. (Contributed by NM, 24-Dec-2015.) (New usage is discouraged.) (Proof modification is discouraged.)

Theoremax12olem4OLD 2014* Obsolete proof of ax12oOLD 2018 as of 30-Jan-2018. Lemma for ax12oOLD 2018. Construct an intermediate equivalent to ax-12 1950 from two instances of ax-12 1950. (Contributed by NM, 24-Dec-2015.) (New usage is discouraged.) (Proof modification is discouraged.)

Theoremax12olem5OLD 2015 Obsolete proof of ax12oOLD 2018 as of 30-Jan-2018. Lemma for ax12oOLD 2018. See ax12olem6OLD 2016 for derivation of ax12oOLD 2018 from the conclusion. (Contributed by NM, 24-Dec-2015.) (New usage is discouraged.) (Proof modification is discouraged.)

Theoremax12olem6OLD 2016* Obsolete proof of ax12oOLD 2018 as of 30-Jan-2018. Lemma for ax12oOLD 2018. Derivation of ax12oOLD 2018 from the hypotheses, without using ax12oOLD 2018. (Contributed by Andrew Salmon, 21-Jul-2011.) (Revised by NM, 24-Dec-2015.) (New usage is discouraged.) (Proof modification is discouraged.)

Theoremax12olem7OLD 2017* Obsolete proof of ax12oOLD 2018 as of 30-Jan-2018. Lemma for ax12oOLD 2018. Derivation of ax12oOLD 2018 from the hypotheses, without using ax12oOLD 2018. (Contributed by NM, 24-Dec-2015.) (New usage is discouraged.) (Proof modification is discouraged.)

Theoremax12oOLD 2018 Obsolete proof of ax12oOLD 2018 as of 30-Jan-2018. (Contributed by NM, 29-Nov-2015.) (Revised by NM, 24-Dec-2015.) (New usage is discouraged.) (Proof modification is discouraged.)

Theoremax12 2019 Derive ax-12 1950 from ax12v 1951 via ax12o 2010. This shows that the weakening in ax12v 1951 is still sufficient for a complete system. (Contributed by NM, 21-Dec-2015.) (Proof shortened by Wolf Lammen, 31-Jan-2018.)

Theoremax12OLD 2020 Obsolete proof of ax12 2019 as of 31-Jan-2018. (Contributed by NM, 21-Dec-2015.) (New usage is discouraged.) (Proof modification is discouraged.)

Theoremdveeq1 2021* Quantifier introduction when one pair of variables is distinct. Revised to be independent of dvelimv 2074. (Contributed by NM, 2-Jan-2002.) (Revised by Wolf Lammen, 29-Apr-2018.)

Theoremax10lem1 2022* Lemma for ax10 2025. Change bound variable. (Contributed by NM, 22-Jul-2015.)

Theoremax10lem2 2023* Lemma for ax10 2025. Change bound variable. (Contributed by NM, 8-Jul-2016.) (Proof shortened by Wolf Lammen, 17-Feb-2018.)

Theoremax10o2 2024 Same as ax10o 2038 but with reversed antecedent. (Contributed by NM, 25-Jul-2015.)

Theoremax10 2025 Derive set.mm's original ax-10 2217 from others. (Contributed by NM, 25-Jul-2015.) (Revised by NM, 7-Nov-2015.) (Proof shortened by Wolf Lammen, 6-Mar-2018.)

Theoremax10lem2OLD 2026* Obsolete proof of a lemma for ax10 2025 as of 17-Feb-2018. Change free variable. (Contributed by NM, 25-Jul-2015.) (New usage is discouraged.) (Proof modification is discouraged.)

Theoremax10lem3OLD 2027* Obsolete proof of a lemma for ax10 2025 as of 17-Feb-2018. Similar to ax-10 2217 but with distinct variables. (Contributed by NM, 25-Jul-2015.) (New usage is discouraged.) (Proof modification is discouraged.)

TheoremdvelimvOLD 2028* Obsolete proof of dvelimv 2074 as of 17-Feb-2018. (Contributed by NM, 25-Jul-2015.) (New usage is discouraged.) (Proof modification is discouraged.)

Theoremdveeq2OLD 2029* Obsolete proof of dveeq2 2077 as of 25-Feb-2018. (Contributed by NM, 2-Jan-2002.) (New usage is discouraged.) (Proof modification is discouraged.)

Theoremax10lem4OLD 2030* Obsolete proof of ax10lem2 2023 as of 17-Feb-2018. (Contributed by NM, 8-Jul-2016.) (New usage is discouraged.) (Proof modification is discouraged. )

Theoremax10lem5OLD 2031* Obsolete proof of ax10o2 2024 as of 17-Feb-2018. (Contributed by NM, 22-Jul-2015.) (New usage is discouraged.) (Proof modification is discouraged.)

Theoremax10OLD 2032 Obsolete proof of ax10 2025 as of 17-Feb-2018. (Contributed by NM, 25-Jul-2015.) (Revised by NM, 7-Nov-2015.) (New usage is discouraged.) (Proof modification is discouraged.)

Theoremax9OLD 2033 Obsolete proof of ax9 1953 as of 4-Feb-2018. (Contributed by NM, 12-Nov-2013.) (Revised by NM, 25-Jul-2015.) (New usage is discouraged.) (Proof modfication is discouraged.)

Theorema9eOLD 2034 Obsolete proof of a9e 1952 as of 4-Feb-2018. (Contributed by NM, 5-Aug-1993.) (New usage is discouraged.) (Proof modfication is discouraged.)

Theoremaecom 2035 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). (Contributed by NM, 5-Aug-1993.)

Theoremaecoms 2036 A commutation rule for identical variable specifiers. (Contributed by NM, 5-Aug-1993.)

Theoremnaecoms 2037 A commutation rule for distinct variable specifiers. (Contributed by NM, 2-Jan-2002.)

Theoremax10o 2038 Show that ax-10o 2216 can be derived from ax-10 2217 in the form of ax10 2025. Normally, ax10o 2038 should be used rather than ax-10o 2216, except by theorems specifically studying the latter's properties. (Contributed by NM, 16-May-2008.) (Proof shortened by Wolf Lammen, 21-Apr-2018.)

Theoremax10oOLD 2039 Obsolete proof of ax10o 2038 as of 21-Apr-2018. (Contributed by NM, 16-May-2008.) (Proof modification is discouraged.) (New usage is discouraged.)

Theoremhbae 2040 All variables are effectively bound in an identical variable specifier. (Contributed by NM, 5-Aug-1993.) (Proof shortened by Wolf Lammen, 21-Apr-2018.)

TheoremhbaeOLD 2041 Obsolete proof of hbae 2040 as of 21-Apr-2018. (Contributed by NM, 5-Aug-1993.) (Proof modification is discouraged.) (New usage is discouraged.)

Theoremnfae 2042 All variables are effectively bound in an identical variable specifier. (Contributed by Mario Carneiro, 11-Aug-2016.)

Theoremhbnae 2043 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 2044 All variables are effectively bound in a distinct variable specifier. (Contributed by Mario Carneiro, 11-Aug-2016.)

Theoremhbnaes 2045 Rule that applies hbnae 2043 to antecedent. (Contributed by NM, 5-Aug-1993.)

Theoremaevlem1 2046* Lemma for aev 2047 and a16g 2048. Change free and bound variables. (Contributed by NM, 22-Jul-2015.) (Proof shortened by Wolf Lammen, 17-Feb-2018.)

Theoremaev 2047* A "distinctor elimination" lemma with no restrictions on variables in the consequent. (Contributed by NM, 8-Nov-2006.)

Theorema16g 2048* Generalization of ax16 2050. (Contributed by NM, 25-Jul-2015.) (Proof shortened by Wolf Lammen, 18-Feb-2018.)

Theorema16gOLD 2049* Obsolete proof of a16g 2048 as of 18-Feb-2018. (Contributed by NM, 25-Jul-2015.) (New usage is discouraged.) (Proof modification is discouraged.)

Theoremax16 2050* Proof of older axiom ax-16 2221. (Contributed by NM, 8-Nov-2006.) (Revised by NM, 22-Sep-2017.)

Theoremax16i 2051* Inference with ax16 2050 as its conclusion. (Contributed by NM, 20-May-2008.) (Proof modification is discouraged.)

Theorema16gb 2052* A generalization of axiom ax-16 2221. (Contributed by NM, 5-Aug-1993.)

Theorema16nf 2053* If dtru 4390 is false, then there is only one element in the universe, so everything satisfies . (Contributed by Mario Carneiro, 7-Oct-2016.)

TheoremnfeqfOLD 2054 Obsolete proof of nfeqf 2009 as of 29-Apr-2018. (Contributed by Mario Carneiro, 6-Oct-2016.) (Proof modification is discouraged.) (New usage is discouraged.)

Theoremdral2 2055 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.) (Revised by Wolf Lammen, 4-Mar-2018.)

Theoremdral2OLD 2056 Obsolete proof of dral2 2055 as of 4-Mar-2018. (Contributed by NM, 27-Feb-2005.) (New usage is discouraged.) (Proof modification is discouraged.)

Theoremdral1 2057 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.) (Proof shortened by Wolf Lammen, 22-Apr-2018.)

Theoremdral1OLD 2058 Obsolete proof of dral1 2057 as of 4-Mar-2018. (Contributed by NM, 24-Nov-1994.) (New usage is discouraged.) (Proof modification is discouraged.)

Theoremdrex1 2059 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 2060 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 2061 Formula-building lemma for use with the Distinctor Reduction Theorem. (Contributed by Mario Carneiro, 4-Oct-2016.)

Theoremdrnf2 2062 Formula-building lemma for use with the Distinctor Reduction Theorem. (Contributed by Mario Carneiro, 4-Oct-2016.) (Proof shortened by Wolf Lammen, 5-May-2018.)

Theoremdrnf2OLD 2063 Obsolete proof of drnf2 2062 as of 5-May-2018. (Contributed by Mario Carneiro, 4-Oct-2016.) (New usage is discouraged.) (Proof modification is discouraged.)

Theoremnfald2 2064 Variation on nfald 1871 which adds the hypothesis that and are distinct in the inner subproof. (Contributed by Mario Carneiro, 8-Oct-2016.)

Theoremnfexd2 2065 Variation on nfexd 1873 which adds the hypothesis that and are distinct in the inner subproof. (Contributed by Mario Carneiro, 8-Oct-2016.)

Theoremexdistrf 2066 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.) (Proof shortened by Wolf Lammen, 14-May-2018.)

TheoremexdistrfOLD 2067 Obsolete proof of exdistrf 2066 as of 14-May-2018. (Contributed by Mario Carneiro, 20-Mar-2013.) (New usage is discouraged.) (Proof modification is discouraged.)

Theoremdvelimf 2068 Version of dvelimv 2074 without any variable restrictions. (Contributed by NM, 1-Oct-2002.) (Revised by Mario Carneiro, 6-Oct-2016.) (Proof shortened by Wolf Lammen, 11-May-2018.)

TheoremdvelimfOLD 2069 Obsolete proof of dvelimf 2068 as of 21-Apr-2018. (Contributed by NM, 1-Oct-2002.) (Revised by Mario Carneiro, 6-Oct-2016.) (Proof modification is discouraged.) (New usage is discouraged.)

Theoremdvelimdf 2070 Deduction form of dvelimf 2068. This version may be useful if we want to avoid ax-17 1626 and use ax-16 2221 instead. (Contributed by NM, 7-Apr-2004.) (Revised by Mario Carneiro, 6-Oct-2016.) (Proof shortened by Wolf Lammen, 11-May-2018.)

Theoremdvelimh 2071 Version of dvelim 2073 without any variable restrictions. (Contributed by NM, 1-Oct-2002.) (Proof shortened by Wolf Lammen, 11-May-2018.)

TheoremdvelimhOLD 2072 Obsolete proof of dvelimh 2071 as of 4-Mar-2018. (Contributed by NM, 1-Oct-2002.) (New usage is discouraged.) (Proof modification is discouraged.)

Theoremdvelim 2073* 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 2070.

Other variants of this theorem are dvelimh 2071 (with no distinct variable restrictions), dvelimhw 1876 (that avoids ax-12 1950), and dvelimALT 2210 (that avoids ax-10 2217). (Contributed by NM, 23-Nov-1994.)

Theoremdvelimv 2074* Similar to dvelim 2073 with first hypothesis replaced by distinct variable condition. (Contributed by NM, 25-Jul-2015.) (Proof shortened by Wolf Lammen, 30-Apr-2018.)

Theoremdvelimnf 2075* Version of dvelim 2073 using "not free" notation. (Contributed by Mario Carneiro, 9-Oct-2016.)

Theoremdveeq1OLD 2076* Obsolete proof of dveeq1 2021 as of 25-Feb-2018. (Contributed by NM, 2-Jan-2002.) (New usage is discouraged.) (Proof modification is discouraged.)

Theoremdveeq2 2077* Quantifier introduction when one pair of variables is distinct. (Contributed by NM, 2-Jan-2002.) (Revised by NM, 20-Jul-2015.)

Theoremax11v2 2078* Recovery of ax-11o 2218 from ax11v 2172. This proof uses ax-10 2217 and ax-11 1761. TODO: figure out if this is useful, or if it should be simplified or eliminated. (Contributed by NM, 2-Feb-2007.) (Proof shortened by Wolf Lammen, 21-Apr-2018.)

Theoremax11v2OLD 2079* Obsolete proof of ax11v2 2078 as of 21-Apr-2018. (Contributed by NM, 2-Feb-2007.) (Proof modification is discouraged.) (New usage is discouraged.)

Theoremax11a2 2080* Derive ax-11o 2218 from a hypothesis in the form of ax-11 1761. ax-10 2217 and ax-11 1761 are used by the proof, but not ax-10o 2216 or ax-11o 2218. TODO: figure out if this is useful, or if it should be simplified or eliminated. (Contributed by NM, 2-Feb-2007.)

Theoremax11o 2081 Derivation of set.mm's original ax-11o 2218 from ax-10 2217 and the shorter ax-11 1761 that has replaced it.

Theorem ax11 2232 shows the reverse derivation of ax-11 1761 from ax-11o 2218.

Normally, ax11o 2081 should be used rather than ax-11o 2218, except by theorems specifically studying the latter's properties. (Contributed by NM, 3-Feb-2007.)

Theoremax11b 2082 A bidirectional version of ax11o 2081. (Contributed by NM, 30-Jun-2006.)

Theoremequvini 2083 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.) (Proof shortened by Wolf Lammen, 7-Apr-2018.)

TheoremequviniOLD 2084 Obsolete proof of equvini 2083 as of 7-Apr-2018. (Contributed by NM, 5-Aug-1993.) (Proof shortened by Andrew Salmon, 25-May-2011.) (Proof modification is discouraged.) (New usage is discouraged.)

Theoremequveli 2085 A variable elimination law for equality with no distinct variable requirements. (Compare equvini 2083.) (Contributed by NM, 1-Mar-2013.) (Proof shortened by Mario Carneiro, 17-Oct-2016.) (Proof shortened by Wolf Lammen, 15-Apr-2018.)

TheoremequveliOLD 2086 Obsolete proof of equveli 2085 as of 15-Apr-2018. (Contributed by NM, 1-Mar-2013.) (Proof shortened by Mario Carneiro, 17-Oct-2016.) (Proof modification is discouraged.) (New usage is discouraged.)

Theoremequvin 2087* A variable introduction law for equality. Lemma 15 of [Monk2] p. 109. (Contributed by NM, 5-Aug-1993.)

Theoremequs45f 2088 Two ways of expressing substitution when is not free in . (Contributed by NM, 25-Apr-2008.) (Revised by Mario Carneiro, 4-Oct-2016.)

Theoremequs5 2089 Lemma used in proofs of substitution properties. (Contributed by NM, 5-Aug-1993.)

Theoremsb2 2090 One direction of a simplified definition of substitution. (Contributed by NM, 5-Aug-1993.)

Theoremstdpc4 2091 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 3173 and rspsbc 3239. (Contributed by NM, 5-Aug-1993.)

Theoremsb3 2092 One direction of a simplified definition of substitution when variables are distinct. (Contributed by NM, 5-Aug-1993.)

Theoremsb4 2093 One direction of a simplified definition of substitution when variables are distinct. (Contributed by NM, 5-Aug-1993.)

Theoremsb4b 2094 Simplified definition of substitution when variables are distinct. (Contributed by NM, 27-May-1997.)

Theoremhbsb2 2095 Bound-variable hypothesis builder for substitution. (Contributed by NM, 5-Aug-1993.)

Theoremnfsb2 2096 Bound-variable hypothesis builder for substitution. (Contributed by Mario Carneiro, 4-Oct-2016.)

Theoremhbsb2a 2097 Special case of a bound-variable hypothesis builder for substitution. (Contributed by NM, 2-Feb-2007.)

Theoremhbsb2e 2098 Special case of a bound-variable hypothesis builder for substitution. (Contributed by NM, 2-Feb-2007.)

Theoremhbsb3 2099 If is not free in , is not free in . (Contributed by NM, 5-Aug-1993.)

Theoremnfs1 2100 If is not free in , is not free in . (Contributed by Mario Carneiro, 11-Aug-2016.)

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