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Theorem List for New Foundations Explorer - 2701-2800   *Has distinct variable group(s)
TypeLabelDescription
Statement
 
Theoremr19.21v 2701* 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 2702 Inference from Theorem 19.21 of [Margaris] p. 90. (Restricted quantifier version.) (Contributed by NM, 16-Feb-2004.)

 F/   &     F/   &       =>   
 
Theoremralrimdv 2703* Inference from Theorem 19.21 of [Margaris] p. 90. (Restricted quantifier version.) (Contributed by NM, 27-May-1998.)
   =>   
 
Theoremralrimdva 2704* Inference from Theorem 19.21 of [Margaris] p. 90. (Restricted quantifier version.) (Contributed by NM, 2-Feb-2008.)
   =>   
 
Theoremralrimivv 2705* Inference from Theorem 19.21 of [Margaris] p. 90. (Restricted quantifier version with double quantification.) (Contributed by NM, 24-Jul-2004.)
   =>   
 
Theoremralrimivva 2706* Inference from Theorem 19.21 of [Margaris] p. 90. (Restricted quantifier version with double quantification.) (Contributed by Jeff Madsen, 19-Jun-2011.)
   =>   
 
Theoremralrimivvva 2707* Inference from Theorem 19.21 of [Margaris] p. 90. (Restricted quantifier version with triple quantification.) (Contributed by Mario Carneiro, 9-Jul-2014.)
   =>   
 
Theoremralrimdvv 2708* Inference from Theorem 19.21 of [Margaris] p. 90. (Restricted quantifier version with double quantification.) (Contributed by NM, 1-Jun-2005.)
   =>   
 
Theoremralrimdvva 2709* Inference from Theorem 19.21 of [Margaris] p. 90. (Restricted quantifier version with double quantification.) (Contributed by NM, 2-Feb-2008.)
   =>   
 
Theoremrgen2 2710* Generalization rule for restricted quantification. (Contributed by NM, 30-May-1999.)
   =>   
 
Theoremrgen3 2711* Generalization rule for restricted quantification. (Contributed by NM, 12-Jan-2008.)
   =>   
 
Theoremr19.21bi 2712 Inference from Theorem 19.21 of [Margaris] p. 90. (Restricted quantifier version.) (Contributed by NM, 20-Nov-1994.)
   =>   
 
Theoremrspec2 2713 Specialization rule for restricted quantification. (Contributed by NM, 20-Nov-1994.)
   =>   
 
Theoremrspec3 2714 Specialization rule for restricted quantification. (Contributed by NM, 20-Nov-1994.)
   =>   
 
Theoremr19.21be 2715 Inference from Theorem 19.21 of [Margaris] p. 90. (Restricted quantifier version.) (Contributed by NM, 21-Nov-1994.)
   =>   
 
Theoremnrex 2716 Inference adding restricted existential quantifier to negated wff. (Contributed by NM, 16-Oct-2003.)
   =>   
 
Theoremnrexdv 2717* Deduction adding restricted existential quantifier to negated wff. (Contributed by NM, 16-Oct-2003.)
   =>   
 
Theoremrexim 2718 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 2719 Inference quantifying both antecedent and consequent. (Contributed by NM, 10-Feb-1997.)
   =>   
 
Theoremreximi2 2720 Inference quantifying both antecedent and consequent, based on Theorem 19.22 of [Margaris] p. 90. (Contributed by NM, 8-Nov-2004.)
   =>   
 
Theoremreximi 2721 Inference quantifying both antecedent and consequent. (Contributed by NM, 18-Oct-1996.)
   =>   
 
Theoremreximdai 2722 Deduction from Theorem 19.22 of [Margaris] p. 90. (Restricted quantifier version.) (Contributed by NM, 31-Aug-1999.)

 F/   &       =>   
 
Theoremreximdv2 2723* Deduction quantifying both antecedent and consequent, based on Theorem 19.22 of [Margaris] p. 90. (Contributed by NM, 17-Sep-2003.)
   =>   
 
Theoremreximdvai 2724* Deduction quantifying both antecedent and consequent, based on Theorem 19.22 of [Margaris] p. 90. (Contributed by NM, 14-Nov-2002.)
   =>   
 
Theoremreximdv 2725* Deduction from Theorem 19.22 of [Margaris] p. 90. (Restricted quantifier version with strong hypothesis.) (Contributed by NM, 24-Jun-1998.)
   =>   
 
Theoremreximdva 2726* Deduction quantifying both antecedent and consequent, based on Theorem 19.22 of [Margaris] p. 90. (Contributed by NM, 22-May-1999.)
   =>   
 
Theoremr19.12 2727* 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 2728 Closed theorem form of r19.23 2729. (Contributed by NM, 4-Mar-2013.) (Revised by Mario Carneiro, 8-Oct-2016.)
 F/
 
Theoremr19.23 2729 Theorem 19.23 of [Margaris] p. 90 with restricted quantifiers. (Contributed by NM, 22-Oct-2010.) (Proof shortened by Mario Carneiro, 8-Oct-2016.)

 F/   =>   
 
Theoremr19.23v 2730* Theorem 19.23 of [Margaris] p. 90 with restricted quantifiers. (Contributed by NM, 31-Aug-1999.)
 
Theoremrexlimi 2731 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.)

 F/   &       =>   
 
Theoremrexlimiv 2732* Inference from Theorem 19.23 of [Margaris] p. 90. (Restricted quantifier version.) (Contributed by NM, 20-Nov-1994.)
   =>   
 
Theoremrexlimiva 2733* Inference from Theorem 19.23 of [Margaris] p. 90 (restricted quantifier version). (Contributed by NM, 18-Dec-2006.)
   =>   
 
Theoremrexlimivw 2734* Weaker version of rexlimiv 2732. (Contributed by FL, 19-Sep-2011.)
   =>   
 
Theoremrexlimd 2735 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.)

 F/   &     F/   &       =>   
 
Theoremrexlimd2 2736 Version of rexlimd 2735 with deduction version of second hypothesis. (Contributed by NM, 21-Jul-2013.) (Revised by Mario Carneiro, 8-Oct-2016.)

 F/   &     F/   &       =>   
 
Theoremrexlimdv 2737* 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 2738* Inference from Theorem 19.23 of [Margaris] p. 90 (restricted quantifier version). (Contributed by NM, 20-Jan-2007.)
   =>   
 
Theoremrexlimdvaa 2739* Inference from Theorem 19.23 of [Margaris] p. 90 (restricted quantifier version). (Contributed by Mario Carneiro, 15-Jun-2016.)
   =>   
 
Theoremrexlimdv3a 2740* Inference from Theorem 19.23 of [Margaris] p. 90 (restricted quantifier version). Frequently-used variant of rexlimdv 2737. (Contributed by NM, 7-Jun-2015.)
   =>   
 
Theoremrexlimdvw 2741* Inference from Theorem 19.23 of [Margaris] p. 90 (restricted quantifier version). (Contributed by NM, 18-Jun-2014.)
   =>   
 
Theoremrexlimddv 2742* Restricted existential elimination rule of natural deduction. (Contributed by Mario Carneiro, 15-Jun-2016.)
   &       =>   
 
Theoremrexlimivv 2743* Inference from Theorem 19.23 of [Margaris] p. 90 (restricted quantifier version). (Contributed by NM, 17-Feb-2004.)
   =>   
 
Theoremrexlimdvv 2744* Inference from Theorem 19.23 of [Margaris] p. 90. (Restricted quantifier version.) (Contributed by NM, 22-Jul-2004.)
   =>   
 
Theoremrexlimdvva 2745* Inference from Theorem 19.23 of [Margaris] p. 90. (Restricted quantifier version.) (Contributed by NM, 18-Jun-2014.)
   =>   
 
Theoremr19.26 2746 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 2747 Theorem 19.26 of [Margaris] p. 90 with 2 restricted quantifiers. (Contributed by NM, 10-Aug-2004.)
 
Theoremr19.26-3 2748 Theorem 19.26 of [Margaris] p. 90 with 3 restricted quantifiers. (Contributed by FL, 22-Nov-2010.)
 
Theoremr19.26m 2749 Theorem 19.26 of [Margaris] p. 90 with mixed quantifiers. (Contributed by NM, 22-Feb-2004.)
 
Theoremralbi 2750 Distribute a restricted universal quantifier over a biconditional. Theorem 19.15 of [Margaris] p. 90 with restricted quantification. (Contributed by NM, 6-Oct-2003.)
 
Theoremralbiim 2751 Split a biconditional and distribute quantifier. (Contributed by NM, 3-Jun-2012.)
 
Theoremr19.27av 2752* Restricted version of one direction of Theorem 19.27 of [Margaris] p. 90. (The other direction doesn't hold when is empty.) (Contributed by NM, 3-Jun-2004.) (Proof shortened by Andrew Salmon, 30-May-2011.)
 
Theoremr19.28av 2753* Restricted version of one direction of Theorem 19.28 of [Margaris] p. 90. (The other direction doesn't hold when is empty.) (Contributed by NM, 2-Apr-2004.)
 
Theoremr19.29 2754 Theorem 19.29 of [Margaris] p. 90 with restricted quantifiers. (Contributed by NM, 31-Aug-1999.) (Proof shortened by Andrew Salmon, 30-May-2011.)
 
Theoremr19.29r 2755 Variation of Theorem 19.29 of [Margaris] p. 90 with restricted quantifiers. (Contributed by NM, 31-Aug-1999.)
 
Theoremr19.30 2756 Theorem 19.30 of [Margaris] p. 90 with restricted quantifiers. (Contributed by Scott Fenton, 25-Feb-2011.)
 
Theoremr19.32v 2757* Theorem 19.32 of [Margaris] p. 90 with restricted quantifiers. (Contributed by NM, 25-Nov-2003.)
 
Theoremr19.35 2758 Restricted quantifier version of Theorem 19.35 of [Margaris] p. 90. (Contributed by NM, 20-Sep-2003.)
 
Theoremr19.36av 2759* One direction of a restricted quantifier version of Theorem 19.36 of [Margaris] p. 90. The other direction doesn't hold when is empty. (Contributed by NM, 22-Oct-2003.)
 
Theoremr19.37 2760 Restricted version of one direction of Theorem 19.37 of [Margaris] p. 90. (The other direction doesn't hold when is empty.) (Contributed by FL, 13-May-2012.) (Revised by Mario Carneiro, 11-Dec-2016.)

 F/   =>   
 
Theoremr19.37av 2761* Restricted version of one direction of Theorem 19.37 of [Margaris] p. 90. (The other direction doesn't hold when is empty.) (Contributed by NM, 2-Apr-2004.)
 
Theoremr19.40 2762 Restricted quantifier version of Theorem 19.40 of [Margaris] p. 90. (Contributed by NM, 2-Apr-2004.)
 
Theoremr19.41 2763 Restricted quantifier version of Theorem 19.41 of [Margaris] p. 90. (Contributed by NM, 1-Nov-2010.)

 F/   =>   
 
Theoremr19.41v 2764* Restricted quantifier version of Theorem 19.41 of [Margaris] p. 90. (Contributed by NM, 17-Dec-2003.)
 
Theoremr19.42v 2765* Restricted version of Theorem 19.42 of [Margaris] p. 90. (Contributed by NM, 27-May-1998.)
 
Theoremr19.43 2766 Restricted version of Theorem 19.43 of [Margaris] p. 90. (Contributed by NM, 27-May-1998.) (Proof shortened by Andrew Salmon, 30-May-2011.)
 
Theoremr19.44av 2767* One direction of a restricted quantifier version of Theorem 19.44 of [Margaris] p. 90. The other direction doesn't hold when is empty. (Contributed by NM, 2-Apr-2004.)
 
Theoremr19.45av 2768* Restricted version of one direction of Theorem 19.45 of [Margaris] p. 90. (The other direction doesn't hold when is empty.) (Contributed by NM, 2-Apr-2004.)
 
Theoremralcomf 2769* Commutation of restricted quantifiers. (Contributed by Mario Carneiro, 14-Oct-2016.)
 F/_   &     F/_   =>   
 
Theoremrexcomf 2770* Commutation of restricted quantifiers. (Contributed by Mario Carneiro, 14-Oct-2016.)
 F/_   &     F/_   =>   
 
Theoremralcom 2771* Commutation of restricted quantifiers. (Contributed by NM, 13-Oct-1999.) (Revised by Mario Carneiro, 14-Oct-2016.)
 
Theoremrexcom 2772* Commutation of restricted quantifiers. (Contributed by NM, 19-Nov-1995.) (Revised by Mario Carneiro, 14-Oct-2016.)
 
Theoremrexcom13 2773* Swap 1st and 3rd restricted existential quantifiers. (Contributed by NM, 8-Apr-2015.)
 
Theoremrexrot4 2774* Rotate existential restricted quantifiers twice. (Contributed by NM, 8-Apr-2015.)
 
Theoremralcom2 2775* Commutation of restricted quantifiers. Note that and needn't be distinct (this makes the proof longer). (Contributed by NM, 24-Nov-1994.) (Proof shortened by Mario Carneiro, 17-Oct-2016.)
 
Theoremralcom3 2776 A commutative law for restricted quantifiers that swaps the domain of the restriction. (Contributed by NM, 22-Feb-2004.)
 
Theoremreean 2777* Rearrange existential quantifiers. (Contributed by NM, 27-Oct-2010.) (Proof shortened by Andrew Salmon, 30-May-2011.)

 F/   &     F/   =>   
 
Theoremreeanv 2778* Rearrange existential quantifiers. (Contributed by NM, 9-May-1999.)
 
Theorem3reeanv 2779* Rearrange three existential quantifiers. (Contributed by Jeff Madsen, 11-Jun-2010.)
 
Theorem2ralor 2780* Distribute quantification over "or". (Contributed by Jeff Madsen, 19-Jun-2010.)
 
Theoremnfreu1 2781 is not free in . (Contributed by NM, 19-Mar-1997.)

 F/
 
Theoremnfrmo1 2782 is not free in . (Contributed by NM, 16-Jun-2017.)

 F/
 
Theoremnfreud 2783 Deduction version of nfreu 2785. (Contributed by NM, 15-Feb-2013.) (Revised by Mario Carneiro, 8-Oct-2016.)

 F/   &     F/_   &     F/   =>     F/
 
Theoremnfrmod 2784 Deduction version of nfrmo 2786. (Contributed by NM, 17-Jun-2017.)

 F/   &     F/_   &     F/   =>     F/
 
Theoremnfreu 2785 Bound-variable hypothesis builder for restricted uniqueness. (Contributed by NM, 30-Oct-2010.) (Revised by Mario Carneiro, 8-Oct-2016.)
 F/_   &     F/   =>    
 F/
 
Theoremnfrmo 2786 Bound-variable hypothesis builder for restricted uniqueness. (Contributed by NM, 16-Jun-2017.)
 F/_   &     F/   =>    
 F/
 
Theoremrabid 2787 An "identity" law of concretion for restricted abstraction. Special case of Definition 2.1 of [Quine] p. 16. (Contributed by NM, 9-Oct-2003.)
 
Theoremrabid2 2788* An "identity" law for restricted class abstraction. (Contributed by NM, 9-Oct-2003.) (Proof shortened by Andrew Salmon, 30-May-2011.)
 
Theoremrabbi 2789 Equivalent wff's correspond to equal restricted class abstractions. Closed theorem form of rabbidva 2850. (Contributed by NM, 25-Nov-2013.)
 
Theoremrabswap 2790 Swap with a membership relation in a restricted class abstraction. (Contributed by NM, 4-Jul-2005.)
 
Theoremnfrab1 2791 The abstraction variable in a restricted class abstraction isn't free. (Contributed by NM, 19-Mar-1997.)
 F/_
 
Theoremnfrab 2792 A variable not free in a wff remains so in a restricted class abstraction. (Contributed by NM, 13-Oct-2003.) (Revised by Mario Carneiro, 9-Oct-2016.)

 F/   &     F/_   =>     F/_
 
Theoremreubida 2793 Formula-building rule for restricted existential quantifier (deduction rule). (Contributed by Mario Carneiro, 19-Nov-2016.)

 F/   &       =>   
 
Theoremreubidva 2794* Formula-building rule for restricted existential quantifier (deduction rule). (Contributed by NM, 13-Nov-2004.)
   =>   
 
Theoremreubidv 2795* Formula-building rule for restricted existential quantifier (deduction rule). (Contributed by NM, 17-Oct-1996.)
   =>   
 
Theoremreubiia 2796 Formula-building rule for restricted existential quantifier (inference rule). (Contributed by NM, 14-Nov-2004.)
   =>   
 
Theoremreubii 2797 Formula-building rule for restricted existential quantifier (inference rule). (Contributed by NM, 22-Oct-1999.)
   =>   
 
Theoremrmobida 2798 Formula-building rule for restricted existential quantifier (deduction rule). (Contributed by NM, 16-Jun-2017.)

 F/   &       =>   
 
Theoremrmobidva 2799* Formula-building rule for restricted existential quantifier (deduction rule). (Contributed by NM, 16-Jun-2017.)
   =>   
 
Theoremrmobidv 2800* Formula-building rule for restricted existential quantifier (deduction rule). (Contributed by NM, 16-Jun-2017.)
   =>   
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