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Theorem List for Metamath Proof Explorer - 2701-2800   *Has distinct variable group(s)
TypeLabelDescription
Statement

Theoremr2alf 2701* Double restricted universal quantification. (Contributed by Mario Carneiro, 14-Oct-2016.)

Theoremr2exf 2702* Double restricted existential quantification. (Contributed by Mario Carneiro, 14-Oct-2016.)

Theoremr2al 2703* Double restricted universal quantification. (Contributed by NM, 19-Nov-1995.)

Theoremr2ex 2704* Double restricted existential quantification. (Contributed by NM, 11-Nov-1995.)

Theorem2ralbida 2705* Formula-building rule for restricted universal quantifier (deduction rule). (Contributed by NM, 24-Feb-2004.)

Theorem2ralbidva 2706* Formula-building rule for restricted universal quantifiers (deduction rule). (Contributed by NM, 4-Mar-1997.)

Theorem2rexbidva 2707* Formula-building rule for restricted existential quantifiers (deduction rule). (Contributed by NM, 15-Dec-2004.)

Theorem2ralbidv 2708* Formula-building rule for restricted universal quantifiers (deduction rule). (Contributed by NM, 28-Jan-2006.) (Revised by Szymon Jaroszewicz, 16-Mar-2007.)

Theorem2rexbidv 2709* Formula-building rule for restricted existential quantifiers (deduction rule). (Contributed by NM, 28-Jan-2006.)

Theoremrexralbidv 2710* Formula-building rule for restricted quantifiers (deduction rule). (Contributed by NM, 28-Jan-2006.)

Theoremralinexa 2711 A transformation of restricted quantifiers and logical connectives. (Contributed by NM, 4-Sep-2005.)

Theoremrexanali 2712 A transformation of restricted quantifiers and logical connectives. (Contributed by NM, 4-Sep-2005.)

Theoremrisset 2713* Two ways to say " belongs to ." (Contributed by NM, 22-Nov-1994.)

Theoremhbral 2714 Bound-variable hypothesis builder for restricted quantification. (Contributed by NM, 1-Sep-1999.) (Revised by David Abernethy, 13-Dec-2009.)

Theoremhbra1 2715 is not free in . (Contributed by NM, 18-Oct-1996.)

Theoremnfra1 2716 is not free in . (Contributed by NM, 18-Oct-1996.) (Revised by Mario Carneiro, 7-Oct-2016.)

Theoremnfrald 2717 Deduction version of nfral 2719. (Contributed by NM, 15-Feb-2013.) (Revised by Mario Carneiro, 7-Oct-2016.)

Theoremnfrexd 2718 Deduction version of nfrex 2721. (Contributed by Mario Carneiro, 14-Oct-2016.)

Theoremnfral 2719 Bound-variable hypothesis builder for restricted quantification. (Contributed by NM, 1-Sep-1999.) (Revised by Mario Carneiro, 7-Oct-2016.)

Theoremnfra2 2720* Similar to Lemma 24 of [Monk2] p. 114, except the quantification of the antecedent is restricted. Derived automatically from hbra2VD 28681. Contributed by Alan Sare 31-Dec-2011. (Contributed by NM, 31-Dec-2011.)

Theoremnfrex 2721 Bound-variable hypothesis builder for restricted quantification. (Contributed by NM, 1-Sep-1999.) (Revised by Mario Carneiro, 7-Oct-2016.)

Theoremnfre1 2722 is not free in . (Contributed by NM, 19-Mar-1997.) (Revised by Mario Carneiro, 7-Oct-2016.)

Theoremr3al 2723* Triple restricted universal quantification. (Contributed by NM, 19-Nov-1995.)

Theoremalral 2724 Universal quantification implies restricted quantification. (Contributed by NM, 20-Oct-2006.)

Theoremrexex 2725 Restricted existence implies existence. (Contributed by NM, 11-Nov-1995.)

Theoremrsp 2726 Restricted specialization. (Contributed by NM, 17-Oct-1996.)

Theoremrspe 2727 Restricted specialization. (Contributed by NM, 12-Oct-1999.)

Theoremrsp2 2728 Restricted specialization. (Contributed by NM, 11-Feb-1997.)

Theoremrsp2e 2729 Restricted specialization. (Contributed by FL, 4-Jun-2012.)

Theoremrspec 2730 Specialization rule for restricted quantification. (Contributed by NM, 19-Nov-1994.)

Theoremrgen 2731 Generalization rule for restricted quantification. (Contributed by NM, 19-Nov-1994.)

Theoremrgen2a 2732* Generalization rule for restricted quantification. Note that and needn't be distinct (and illustrates the use of dvelim 2066). (Contributed by NM, 23-Nov-1994.) (Proof shortened by Andrew Salmon, 25-May-2011.) (Proof modification is discouraged.

Theoremrgenw 2733 Generalization rule for restricted quantification. (Contributed by NM, 18-Jun-2014.)

Theoremrgen2w 2734 Generalization rule for restricted quantification. Note that and needn't be distinct. (Contributed by NM, 18-Jun-2014.)

Theoremmprg 2735 Modus ponens combined with restricted generalization. (Contributed by NM, 10-Aug-2004.)

Theoremmprgbir 2736 Modus ponens on biconditional combined with restricted generalization. (Contributed by NM, 21-Mar-2004.)

Theoremralim 2737 Distribution of restricted quantification over implication. (Contributed by NM, 9-Feb-1997.)

Theoremralimi2 2738 Inference quantifying both antecedent and consequent. (Contributed by NM, 22-Feb-2004.)

Theoremralimia 2739 Inference quantifying both antecedent and consequent. (Contributed by NM, 19-Jul-1996.)

Theoremralimiaa 2740 Inference quantifying both antecedent and consequent. (Contributed by NM, 4-Aug-2007.)

Theoremralimi 2741 Inference quantifying both antecedent and consequent, with strong hypothesis. (Contributed by NM, 4-Mar-1997.)

Theoremral2imi 2742 Inference quantifying antecedent, nested antecedent, and consequent, with a strong hypothesis. (Contributed by NM, 19-Dec-2006.)

Theoremralimdaa 2743 Deduction quantifying both antecedent and consequent, based on Theorem 19.20 of [Margaris] p. 90. (Contributed by NM, 22-Sep-2003.)

Theoremralimdva 2744* Deduction quantifying both antecedent and consequent, based on Theorem 19.20 of [Margaris] p. 90. (Contributed by NM, 22-May-1999.)

Theoremralimdv 2745* Deduction quantifying both antecedent and consequent, based on Theorem 19.20 of [Margaris] p. 90. (Contributed by NM, 8-Oct-2003.)

Theoremralimdv2 2746* Inference quantifying both antecedent and consequent. (Contributed by NM, 1-Feb-2005.)

Theoremralrimi 2747 Inference from Theorem 19.21 of [Margaris] p. 90 (restricted quantifier version). (Contributed by NM, 10-Oct-1999.)

Theoremralrimiv 2748* Inference from Theorem 19.21 of [Margaris] p. 90. (Restricted quantifier version.) (Contributed by NM, 22-Nov-1994.)

Theoremralrimiva 2749* Inference from Theorem 19.21 of [Margaris] p. 90. (Restricted quantifier version.) (Contributed by NM, 2-Jan-2006.)

Theoremralrimivw 2750* Inference from Theorem 19.21 of [Margaris] p. 90. (Restricted quantifier version.) (Contributed by NM, 18-Jun-2014.)

Theoremr19.21t 2751 Theorem 19.21 of [Margaris] p. 90 with restricted quantifiers (closed theorem version). (Contributed by NM, 1-Mar-2008.)

Theoremr19.21 2752 Theorem 19.21 of [Margaris] p. 90 with restricted quantifiers. (Contributed by Scott Fenton, 30-Mar-2011.)

Theoremr19.21v 2753* Theorem 19.21 of [Margaris] p. 90 with restricted quantifiers. (Contributed by NM, 15-Oct-2003.) (Proof shortened by Andrew Salmon, 30-May-2011.)

Theoremralrimd 2754 Inference from Theorem 19.21 of [Margaris] p. 90. (Restricted quantifier version.) (Contributed by NM, 16-Feb-2004.)

Theoremralrimdv 2755* Inference from Theorem 19.21 of [Margaris] p. 90. (Restricted quantifier version.) (Contributed by NM, 27-May-1998.)

Theoremralrimdva 2756* Inference from Theorem 19.21 of [Margaris] p. 90. (Restricted quantifier version.) (Contributed by NM, 2-Feb-2008.)

Theoremralrimivv 2757* Inference from Theorem 19.21 of [Margaris] p. 90. (Restricted quantifier version with double quantification.) (Contributed by NM, 24-Jul-2004.)

Theoremralrimivva 2758* Inference from Theorem 19.21 of [Margaris] p. 90. (Restricted quantifier version with double quantification.) (Contributed by Jeff Madsen, 19-Jun-2011.)

Theoremralrimivvva 2759* Inference from Theorem 19.21 of [Margaris] p. 90. (Restricted quantifier version with triple quantification.) (Contributed by Mario Carneiro, 9-Jul-2014.)

Theoremralrimdvv 2760* Inference from Theorem 19.21 of [Margaris] p. 90. (Restricted quantifier version with double quantification.) (Contributed by NM, 1-Jun-2005.)

Theoremralrimdvva 2761* Inference from Theorem 19.21 of [Margaris] p. 90. (Restricted quantifier version with double quantification.) (Contributed by NM, 2-Feb-2008.)

Theoremrgen2 2762* Generalization rule for restricted quantification. (Contributed by NM, 30-May-1999.)

Theoremrgen3 2763* Generalization rule for restricted quantification. (Contributed by NM, 12-Jan-2008.)

Theoremr19.21bi 2764 Inference from Theorem 19.21 of [Margaris] p. 90. (Restricted quantifier version.) (Contributed by NM, 20-Nov-1994.)

Theoremrspec2 2765 Specialization rule for restricted quantification. (Contributed by NM, 20-Nov-1994.)

Theoremrspec3 2766 Specialization rule for restricted quantification. (Contributed by NM, 20-Nov-1994.)

Theoremr19.21be 2767 Inference from Theorem 19.21 of [Margaris] p. 90. (Restricted quantifier version.) (Contributed by NM, 21-Nov-1994.)

Theoremnrex 2768 Inference adding restricted existential quantifier to negated wff. (Contributed by NM, 16-Oct-2003.)

Theoremnrexdv 2769* Deduction adding restricted existential quantifier to negated wff. (Contributed by NM, 16-Oct-2003.)

Theoremrexim 2770 Theorem 19.22 of [Margaris] p. 90. (Restricted quantifier version.) (Contributed by NM, 22-Nov-1994.) (Proof shortened by Andrew Salmon, 30-May-2011.)

Theoremreximia 2771 Inference quantifying both antecedent and consequent. (Contributed by NM, 10-Feb-1997.)

Theoremreximi2 2772 Inference quantifying both antecedent and consequent, based on Theorem 19.22 of [Margaris] p. 90. (Contributed by NM, 8-Nov-2004.)

Theoremreximi 2773 Inference quantifying both antecedent and consequent. (Contributed by NM, 18-Oct-1996.)

Theoremreximdai 2774 Deduction from Theorem 19.22 of [Margaris] p. 90. (Restricted quantifier version.) (Contributed by NM, 31-Aug-1999.)

Theoremreximdv2 2775* Deduction quantifying both antecedent and consequent, based on Theorem 19.22 of [Margaris] p. 90. (Contributed by NM, 17-Sep-2003.)

Theoremreximdvai 2776* Deduction quantifying both antecedent and consequent, based on Theorem 19.22 of [Margaris] p. 90. (Contributed by NM, 14-Nov-2002.)

Theoremreximdv 2777* Deduction from Theorem 19.22 of [Margaris] p. 90. (Restricted quantifier version with strong hypothesis.) (Contributed by NM, 24-Jun-1998.)

Theoremreximdva 2778* Deduction quantifying both antecedent and consequent, based on Theorem 19.22 of [Margaris] p. 90. (Contributed by NM, 22-May-1999.)

Theoremr19.12 2779* Theorem 19.12 of [Margaris] p. 89 with restricted quantifiers. (Contributed by NM, 15-Oct-2003.) (Proof shortened by Andrew Salmon, 30-May-2011.)

Theoremr19.23t 2780 Closed theorem form of r19.23 2781. (Contributed by NM, 4-Mar-2013.) (Revised by Mario Carneiro, 8-Oct-2016.)

Theoremr19.23 2781 Theorem 19.23 of [Margaris] p. 90 with restricted quantifiers. (Contributed by NM, 22-Oct-2010.) (Proof shortened by Mario Carneiro, 8-Oct-2016.)

Theoremr19.23v 2782* Theorem 19.23 of [Margaris] p. 90 with restricted quantifiers. (Contributed by NM, 31-Aug-1999.)

Theoremrexlimi 2783 Inference from Theorem 19.21 of [Margaris] p. 90. (Restricted quantifier version.) (Contributed by NM, 30-Nov-2003.) (Proof shortened by Andrew Salmon, 30-May-2011.)

Theoremrexlimiv 2784* Inference from Theorem 19.23 of [Margaris] p. 90. (Restricted quantifier version.) (Contributed by NM, 20-Nov-1994.)

Theoremrexlimiva 2785* Inference from Theorem 19.23 of [Margaris] p. 90 (restricted quantifier version). (Contributed by NM, 18-Dec-2006.)

Theoremrexlimivw 2786* Weaker version of rexlimiv 2784. (Contributed by FL, 19-Sep-2011.)

Theoremrexlimd 2787 Deduction from Theorem 19.23 of [Margaris] p. 90 (restricted quantifier version). (Contributed by NM, 27-May-1998.) (Proof shortened by Andrew Salmon, 30-May-2011.)

Theoremrexlimd2 2788 Version of rexlimd 2787 with deduction version of second hypothesis. (Contributed by NM, 21-Jul-2013.) (Revised by Mario Carneiro, 8-Oct-2016.)

Theoremrexlimdv 2789* Inference from Theorem 19.23 of [Margaris] p. 90 (restricted quantifier version). (Contributed by NM, 14-Nov-2002.) (Proof shortened by Eric Schmidt, 22-Dec-2006.)

Theoremrexlimdva 2790* Inference from Theorem 19.23 of [Margaris] p. 90 (restricted quantifier version). (Contributed by NM, 20-Jan-2007.)

Theoremrexlimdvaa 2791* Inference from Theorem 19.23 of [Margaris] p. 90 (restricted quantifier version). (Contributed by Mario Carneiro, 15-Jun-2016.)

Theoremrexlimdv3a 2792* Inference from Theorem 19.23 of [Margaris] p. 90 (restricted quantifier version). Frequently-used variant of rexlimdv 2789. (Contributed by NM, 7-Jun-2015.)

Theoremrexlimdvw 2793* Inference from Theorem 19.23 of [Margaris] p. 90 (restricted quantifier version). (Contributed by NM, 18-Jun-2014.)

Theoremrexlimddv 2794* Restricted existential elimination rule of natural deduction. (Contributed by Mario Carneiro, 15-Jun-2016.)

Theoremrexlimivv 2795* Inference from Theorem 19.23 of [Margaris] p. 90 (restricted quantifier version). (Contributed by NM, 17-Feb-2004.)

Theoremrexlimdvv 2796* Inference from Theorem 19.23 of [Margaris] p. 90. (Restricted quantifier version.) (Contributed by NM, 22-Jul-2004.)

Theoremrexlimdvva 2797* Inference from Theorem 19.23 of [Margaris] p. 90. (Restricted quantifier version.) (Contributed by NM, 18-Jun-2014.)

Theoremr19.26 2798 Theorem 19.26 of [Margaris] p. 90 with restricted quantifiers. (Contributed by NM, 28-Jan-1997.) (Proof shortened by Andrew Salmon, 30-May-2011.)

Theoremr19.26-2 2799 Theorem 19.26 of [Margaris] p. 90 with 2 restricted quantifiers. (Contributed by NM, 10-Aug-2004.)

Theoremr19.26-3 2800 Theorem 19.26 of [Margaris] p. 90 with 3 restricted quantifiers. (Contributed by FL, 22-Nov-2010.)

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