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Theorem List for Metamath Proof Explorer - 2801-2900   *Has distinct variable group(s)
TypeLabelDescription
Statement

Theoremcla4gf 2801 Rule of specialization, using implicit substitution. Compare Theorem 7.3 of [Quine] p. 44. (Contributed by NM, 2-Feb-1997.) (Revised by Andrew Salmon, 12-Aug-2011.)

Theoremcla4egf 2802 Existential specialization, using implicit substitution. (Contributed by NM, 2-Feb-1997.)

Theoremcla4imdv 2803* Restricted specialization, using implicit substitution. (Contributed by Mario Carneiro, 4-Jan-2017.)

Theoremcla4imedv 2804* Restricted existential specialization, using implicit substitution. (Contributed by Mario Carneiro, 4-Jan-2017.)

Theoremcla4gv 2805* Rule of specialization, using implicit substitution. Compare Theorem 7.3 of [Quine] p. 44. (Contributed by NM, 22-Jun-1994.)

Theoremcla4egv 2806* Existential specialization, using implicit substitution. (Contributed by NM, 14-Aug-1994.)

Theoremcla42egv 2807* Existential specialization with 2 quantifiers, using implicit substitution. (Contributed by NM, 3-Aug-1995.)

Theoremcla42gv 2808* Specialization with 2 quantifiers, using implicit substitution. (Contributed by NM, 27-Apr-2004.)

Theoremcla43egv 2809* Existential specialization with 3 quantifiers, using implicit substitution. (Contributed by NM, 12-May-2008.)

Theoremcla43gv 2810* Specialization with 3 quantifiers, using implicit substitution. (Contributed by NM, 12-May-2008.)

Theoremcla4v 2811* Rule of specialization, using implicit substitution. (Contributed by NM, 22-Jun-1994.)

Theoremcla4ev 2812* Existential specialization, using implicit substitution. (Contributed by NM, 31-Dec-1993.) (Proof shortened by Eric Schmidt, 22-Dec-2006.)

Theoremcla42ev 2813* Existential specialization, using implicit substitution. (Contributed by NM, 3-Aug-1995.)

Theoremrcla4t 2814* A closed version of rcla4 2815. (Contributed by Andrew Salmon, 6-Jun-2011.)

Theoremrcla4 2815* Restricted specialization, using implicit substitution. (Contributed by NM, 19-Apr-2005.) (Revised by Mario Carneiro, 11-Oct-2016.)

Theoremrcla4e 2816* Restricted existential specialization, using implicit substitution. (Contributed by NM, 26-May-1998.) (Revised by Mario Carneiro, 11-Oct-2016.)

Theoremrcla4v 2817* Restricted specialization, using implicit substitution. (Contributed by NM, 26-May-1998.)

Theoremrcla4cv 2818* Restricted specialization, using implicit substitution. (Contributed by NM, 2-Feb-2006.)

Theoremrcla4va 2819* Restricted specialization, using implicit substitution. (Contributed by NM, 13-Sep-2005.)

Theoremrcla4cva 2820* Restricted specialization, using implicit substitution. (Contributed by NM, 26-Jul-2006.) (Proof shortened by Andrew Salmon, 8-Jun-2011.)

Theoremrcla4ev 2821* Restricted existential specialization, using implicit substitution. (Contributed by NM, 26-May-1998.)

Theoremrcla4imdv 2822* Restricted specialization, using implicit substitution. (Contributed by Mario Carneiro, 4-Jan-2017.)

Theoremrcla4imedv 2823* Restricted existential specialization, using implicit substitution. (Contributed by Mario Carneiro, 4-Jan-2017.)

Theoremrcla4dv 2824* Restricted specialization, using implicit substitution. (Contributed by NM, 17-Feb-2007.) (Revised by Mario Carneiro, 4-Jan-2017.)

Theoremrcla4edv 2825* Restricted existential specialization, using implicit substitution. (Contributed by FL, 17-Apr-2007.) (Revised by Mario Carneiro, 4-Jan-2017.)

Theoremrcla42 2826* 2-variable restricted specialization, using implicit substitution. (Contributed by NM, 9-Nov-2012.)

Theoremrcla42v 2827* 2-variable restricted specialization, using implicit substitution. (Contributed by NM, 13-Sep-1999.)

Theoremrcla42va 2828* 2-variable restricted specialization, using implicit substitution. (Contributed by NM, 18-Jun-2014.)

Theoremrcla42ev 2829* 2-variable restricted existential specialization, using implicit substitution. (Contributed by NM, 16-Oct-1999.)

Theoremrcla43v 2830* 3-variable restricted specialization, using implicit substitution. (Contributed by NM, 10-May-2005.)

Theoremrcla43ev 2831* 3-variable restricted existentional specialization, using implicit substitution. (Contributed by NM, 25-Jul-2012.)

Theoremeqvinc 2832* A variable introduction law for class equality. (Contributed by NM, 14-Apr-1995.) (Proof shortened by Andrew Salmon, 8-Jun-2011.)

Theoremeqvincf 2833 A variable introduction law for class equality, using bound-variable hypotheses instead of distinct variable conditions. (Contributed by NM, 14-Sep-2003.)

Theoremalexeq 2834* Two ways to express substitution of for in . (Contributed by NM, 2-Mar-1995.)

Theoremceqex 2835* Equality implies equivalence with substitution. (Contributed by NM, 2-Mar-1995.)

Theoremceqsexg 2836* A representation of explicit substitution of a class for a variable, inferred from an implicit substitution hypothesis. (Contributed by NM, 11-Oct-2004.)

Theoremceqsexgv 2837* Elimination of an existential quantifier, using implicit substitution. (Contributed by NM, 29-Dec-1996.)

Theoremceqsrexv 2838* Elimination of a restricted existential quantifier, using implicit substitution. (Contributed by NM, 30-Apr-2004.)

Theoremceqsrexbv 2839* Elimination of a restricted existential quantifier, using implicit substitution. (Contributed by Mario Carneiro, 14-Mar-2014.)

Theoremceqsrex2v 2840* Elimination of a restricted existential quantifier, using implicit substitution. (Contributed by NM, 29-Oct-2005.)

Theoremclel2 2841* An alternate definition of class membership when the class is a set. (Contributed by NM, 18-Aug-1993.)

Theoremclel3g 2842* An alternate definition of class membership when the class is a set. (Contributed by NM, 13-Aug-2005.)

Theoremclel3 2843* An alternate definition of class membership when the class is a set. (Contributed by NM, 18-Aug-1993.)

Theoremclel4 2844* An alternate definition of class membership when the class is a set. (Contributed by NM, 18-Aug-1993.)

Theorempm13.183 2845* Compare theorem *13.183 in [WhiteheadRussell] p. 178. Only is required to be a set. (Contributed by Andrew Salmon, 3-Jun-2011.)

Theoremrr19.3v 2846* Restricted quantifier version of Theorem 19.3 of [Margaris] p. 89. We don't need the non-empty class condition of r19.3rzv 3453 when there is an outer quantifier. (Contributed by NM, 25-Oct-2012.)

Theoremrr19.28v 2847* Restricted quantifier version of Theorem 19.28 of [Margaris] p. 90. We don't need the non-empty class condition of r19.28zv 3455 when there is an outer quantifier. (Contributed by NM, 29-Oct-2012.)

Theoremelabgt 2848* Membership in a class abstraction, using implicit substitution. (Closed theorem version of elabg 2852.) (Contributed by NM, 7-Nov-2005.) (Proof shortened by Andrew Salmon, 8-Jun-2011.)

Theoremelabgf 2849 Membership in a class abstraction, using implicit substitution. Compare Theorem 6.13 of [Quine] p. 44. This version has bound-variable hypotheses in place of distinct variable restrictions. (Contributed by NM, 21-Sep-2003.) (Revised by Mario Carneiro, 12-Oct-2016.)

Theoremelabf 2850* Membership in a class abstraction, using implicit substitution. (Contributed by NM, 1-Aug-1994.) (Revised by Mario Carneiro, 12-Oct-2016.)

Theoremelab 2851* Membership in a class abstraction, using implicit substitution. Compare Theorem 6.13 of [Quine] p. 44. (Contributed by NM, 1-Aug-1994.)

Theoremelabg 2852* Membership in a class abstraction, using implicit substitution. Compare Theorem 6.13 of [Quine] p. 44. (Contributed by NM, 14-Apr-1995.)

Theoremelab2g 2853* Membership in a class abstraction, using implicit substitution. (Contributed by NM, 13-Sep-1995.)

Theoremelab2 2854* Membership in a class abstraction, using implicit substitution. (Contributed by NM, 13-Sep-1995.)

Theoremelab4g 2855* Membership in a class abstraction, using implicit substitution. (Contributed by NM, 17-Oct-2012.)

Theoremelab3gf 2856 Membership in a class abstraction, with a weaker antecedent than elabgf 2849. (Contributed by NM, 6-Sep-2011.)

Theoremelab3g 2857* Membership in a class abstraction, with a weaker antecedent than elabg 2852. (Contributed by NM, 29-Aug-2006.)

Theoremelab3 2858* Membership in a class abstraction using implicit substitution. (Contributed by NM, 10-Nov-2000.)

Theoremelrabf 2859 Membership in a restricted class abstraction, using implicit substitution. This version has bound-variable hypotheses in place of distinct variable restrictions. (Contributed by NM, 21-Sep-2003.)

Theoremelrab 2860* Membership in a restricted class abstraction, using implicit substitution. (Contributed by NM, 21-May-1999.)

Theoremelrab3 2861* Membership in a restricted class abstraction, using implicit substitution. (Contributed by NM, 5-Oct-2006.)

Theoremelrab2 2862* Membership in a class abstraction, using implicit substitution. (Contributed by NM, 2-Nov-2006.)

Theoremralab 2863* Universal quantification over a class abstraction. (Contributed by Jeff Madsen, 10-Jun-2010.)

Theoremralrab 2864* Universal quantification over a restricted class abstraction. (Contributed by Jeff Madsen, 10-Jun-2010.)

Theoremrexab 2865* Existential quantification over a class abstraction. (Contributed by Mario Carneiro, 23-Jan-2014.) (Revised by Mario Carneiro, 3-Sep-2015.)

Theoremrexrab 2866* Existential quantification over a class abstraction. (Contributed by Jeff Madsen, 17-Jun-2011.) (Revised by Mario Carneiro, 3-Sep-2015.)

Theoremralab2 2867* Universal quantification over a class abstraction. (Contributed by Mario Carneiro, 3-Sep-2015.)

Theoremralrab2 2868* Universal quantification over a restricted class abstraction. (Contributed by Mario Carneiro, 3-Sep-2015.)

Theoremrexab2 2869* Existential quantification over a class abstraction. (Contributed by Mario Carneiro, 3-Sep-2015.)

Theoremrexrab2 2870* Existential quantification over a class abstraction. (Contributed by Mario Carneiro, 3-Sep-2015.)

Theoremabidnf 2871* Identity used to create closed-form versions of bound-variable hypothesis builders for class expressions. (Contributed by NM, 10-Nov-2005.) (Proof shortened by Mario Carneiro, 12-Oct-2016.)

Theoremdedhb 2872* A deduction theorem for converting the inference => into a closed theorem. Use nfa1 1719 and nfab 2389 to eliminate the hypothesis of the substitution instance of the inference. For converting the inference form into a deduction form, abidnf 2871 is useful. (Contributed by NM, 8-Dec-2006.)

Theoremeqeu 2873* A condition which implies existential uniqueness. (Contributed by Jeff Hankins, 8-Sep-2009.)

Theoremeueq 2874* Equality has existential uniqueness. (Contributed by NM, 25-Nov-1994.)

Theoremeueq1 2875* Equality has existential uniqueness. (Contributed by NM, 5-Apr-1995.)

Theoremeueq2 2876* Equality has existential uniqueness (split into 2 cases). (Contributed by NM, 5-Apr-1995.)

Theoremeueq3 2877* Equality has existential uniqueness (split into 3 cases). (Contributed by NM, 5-Apr-1995.) (Proof shortened by Mario Carneiro, 28-Sep-2015.)

Theoremmoeq 2878* There is at most one set equal to a class. (Contributed by NM, 8-Mar-1995.)

Theoremmoeq3 2879* "At most one" property of equality (split into 3 cases). (The first 2 hypotheses could be eliminated with longer proof.) (Contributed by NM, 23-Apr-1995.)

Theoremmosub 2880* "At most one" remains true after substitution. (Contributed by NM, 9-Mar-1995.)

Theoremmo2icl 2881* Theorem for inferring "at most one." (Contributed by NM, 17-Oct-1996.)

Theoremmob2 2882* Consequence of "at most one." (Contributed by NM, 2-Jan-2015.)

Theoremmoi2 2883* Consequence of "at most one." (Contributed by NM, 29-Jun-2008.)

Theoremmob 2884* Equality implied by "at most one." (Contributed by NM, 18-Feb-2006.)

Theoremmoi 2885* Equality implied by "at most one." (Contributed by NM, 18-Feb-2006.)

Theoremmorex 2886* Derive membership from uniqueness. (Contributed by Jeff Madsen, 2-Sep-2009.)

Theoremeuxfr2 2887* Transfer existential uniqueness from a variable to another variable contained in expression . (Contributed by NM, 14-Nov-2004.)

Theoremeuxfr 2888* Transfer existential uniqueness from a variable to another variable contained in expression . (Contributed by NM, 14-Nov-2004.)

Theoremeuind 2889* Existential uniqueness via an indirect equality. (Contributed by NM, 11-Oct-2010.)

Theoremreurex 2890 Restricted unique existence implies restricted existence. (Contributed by NM, 19-Aug-1999.)

Theoremreu5 2891 Restricted uniqueness in terms of "at most one." (Contributed by NM, 23-May-1999.)

Theoremreu2 2892* A way to express restricted uniqueness. (Contributed by NM, 22-Nov-1994.)

Theoremreu6 2893* A way to express restricted uniqueness. (Contributed by NM, 20-Oct-2006.)

Theoremreu3 2894* A way to express restricted uniqueness. (Contributed by NM, 24-Oct-2006.)

Theoremreu6i 2895* A condition which implies existential uniqueness. (Contributed by Mario Carneiro, 2-Oct-2015.)

Theoremeqreu 2896* A condition which implies existential uniqueness. (Contributed by Mario Carneiro, 2-Oct-2015.)

Theoremrmo4 2897* Restricted "at most one" using implicit substitution. (Contributed by NM, 24-Oct-2006.)

Theoremreu4 2898* Restricted uniqueness using implicit substitution. (Contributed by NM, 23-Nov-1994.)

Theoremreu7 2899* Restricted uniqueness using implicit substitution. (Contributed by NM, 24-Oct-2006.)

Theoremreu8 2900* Restricted uniqueness using implicit substitution. (Contributed by NM, 24-Oct-2006.)

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