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Theorem List for Metamath Proof Explorer - 2601-2700   *Has distinct variable group(s)
TypeLabelDescription
Statement

Theoremalral 2601 Universal quantification implies restricted quantification. (Contributed by NM, 20-Oct-2006.)

Theoremrexex 2602 Restricted existence implies existence. (Contributed by NM, 11-Nov-1995.)

Theoremrsp 2603 Restricted specialization. (Contributed by NM, 17-Oct-1996.)

Theoremrspe 2604 Restricted specialization. (Contributed by NM, 12-Oct-1999.)

Theoremrsp2 2605 Restricted specialization. (Contributed by NM, 11-Feb-1997.)

Theoremrsp2e 2606 Restricted specialization. (Contributed by FL, 4-Jun-2012.)

Theoremrspec 2607 Specialization rule for restricted quantification. (Contributed by NM, 19-Nov-1994.)

Theoremrgen 2608 Generalization rule for restricted quantification. (Contributed by NM, 19-Nov-1994.)

Theoremrgen2a 2609* Generalization rule for restricted quantification. Note that and needn't be distinct (and illustrates the use of dvelim 1956). (Contributed by NM, 23-Nov-1994.) (Proof shortened by Andrew Salmon, 25-May-2011.) (Proof modification is discouraged.

Theoremrgenw 2610 Generalization rule for restricted quantification. (Contributed by NM, 18-Jun-2014.)

Theoremrgen2w 2611 Generalization rule for restricted quantification. Note that and needn't be distinct. (Contributed by NM, 18-Jun-2014.)

Theoremmprg 2612 Modus ponens combined with restricted generalization. (Contributed by NM, 10-Aug-2004.)

Theoremmprgbir 2613 Modus ponens on biconditional combined with restricted generalization. (Contributed by NM, 21-Mar-2004.)

Theoremralim 2614 Distribution of restricted quantification over implication. (Contributed by NM, 9-Feb-1997.)

Theoremralimi2 2615 Inference quantifying both antecedent and consequent. (Contributed by NM, 22-Feb-2004.)

Theoremralimia 2616 Inference quantifying both antecedent and consequent. (Contributed by NM, 19-Jul-1996.)

Theoremralimiaa 2617 Inference quantifying both antecedent and consequent. (Contributed by NM, 4-Aug-2007.)

Theoremralimi 2618 Inference quantifying both antecedent and consequent, with strong hypothesis. (Contributed by NM, 4-Mar-1997.)

Theoremral2imi 2619 Inference quantifying antecedent, nested antecedent, and consequent, with a strong hypothesis. (Contributed by NM, 19-Dec-2006.)

Theoremralimdaa 2620 Deduction quantifying both antecedent and consequent, based on Theorem 19.20 of [Margaris] p. 90. (Contributed by NM, 22-Sep-2003.)

Theoremralimdva 2621* Deduction quantifying both antecedent and consequent, based on Theorem 19.20 of [Margaris] p. 90. (Contributed by NM, 22-May-1999.)

Theoremralimdv 2622* Deduction quantifying both antecedent and consequent, based on Theorem 19.20 of [Margaris] p. 90. (Contributed by NM, 8-Oct-2003.)

Theoremralimdv2 2623* Inference quantifying both antecedent and consequent. (Contributed by NM, 1-Feb-2005.)

Theoremralrimi 2624 Inference from Theorem 19.21 of [Margaris] p. 90 (restricted quantifier version). (Contributed by NM, 10-Oct-1999.)

Theoremralrimiv 2625* Inference from Theorem 19.21 of [Margaris] p. 90. (Restricted quantifier version.) (Contributed by NM, 22-Nov-1994.)

Theoremralrimiva 2626* Inference from Theorem 19.21 of [Margaris] p. 90. (Restricted quantifier version.) (Contributed by NM, 2-Jan-2006.)

Theoremralrimivw 2627* Inference from Theorem 19.21 of [Margaris] p. 90. (Restricted quantifier version.) (Contributed by NM, 18-Jun-2014.)

Theoremr19.21t 2628 Theorem 19.21 of [Margaris] p. 90 with restricted quantifiers (closed theorem version). (Contributed by NM, 1-Mar-2008.)

Theoremr19.21 2629 Theorem 19.21 of [Margaris] p. 90 with restricted quantifiers. (Contributed by Scott Fenton, 30-Mar-2011.)

Theoremr19.21v 2630* 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 2631 Inference from Theorem 19.21 of [Margaris] p. 90. (Restricted quantifier version.) (Contributed by NM, 16-Feb-2004.)

Theoremralrimdv 2632* Inference from Theorem 19.21 of [Margaris] p. 90. (Restricted quantifier version.) (Contributed by NM, 27-May-1998.)

Theoremralrimdva 2633* Inference from Theorem 19.21 of [Margaris] p. 90. (Restricted quantifier version.) (Contributed by NM, 2-Feb-2008.)

Theoremralrimivv 2634* Inference from Theorem 19.21 of [Margaris] p. 90. (Restricted quantifier version with double quantification.) (Contributed by NM, 24-Jul-2004.)

Theoremralrimivva 2635* Inference from Theorem 19.21 of [Margaris] p. 90. (Restricted quantifier version with double quantification.) (Contributed by Jeff Madsen, 19-Jun-2011.)

Theoremralrimivvva 2636* Inference from Theorem 19.21 of [Margaris] p. 90. (Restricted quantifier version with triple quantification.) (Contributed by Mario Carneiro, 9-Jul-2014.)

Theoremralrimdvv 2637* Inference from Theorem 19.21 of [Margaris] p. 90. (Restricted quantifier version with double quantification.) (Contributed by NM, 1-Jun-2005.)

Theoremralrimdvva 2638* Inference from Theorem 19.21 of [Margaris] p. 90. (Restricted quantifier version with double quantification.) (Contributed by NM, 2-Feb-2008.)

Theoremrgen2 2639* Generalization rule for restricted quantification. (Contributed by NM, 30-May-1999.)

Theoremrgen3 2640* Generalization rule for restricted quantification. (Contributed by NM, 12-Jan-2008.)

Theoremr19.21bi 2641 Inference from Theorem 19.21 of [Margaris] p. 90. (Restricted quantifier version.) (Contributed by NM, 20-Nov-1994.)

Theoremrspec2 2642 Specialization rule for restricted quantification. (Contributed by NM, 20-Nov-1994.)

Theoremrspec3 2643 Specialization rule for restricted quantification. (Contributed by NM, 20-Nov-1994.)

Theoremr19.21be 2644 Inference from Theorem 19.21 of [Margaris] p. 90. (Restricted quantifier version.) (Contributed by NM, 21-Nov-1994.)

Theoremnrex 2645 Inference adding restricted existential quantifier to negated wff. (Contributed by NM, 16-Oct-2003.)

Theoremnrexdv 2646* Deduction adding restricted existential quantifier to negated wff. (Contributed by NM, 16-Oct-2003.)

Theoremrexim 2647 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 2648 Inference quantifying both antecedent and consequent. (Contributed by NM, 10-Feb-1997.)

Theoremreximi2 2649 Inference quantifying both antecedent and consequent, based on Theorem 19.22 of [Margaris] p. 90. (Contributed by NM, 8-Nov-2004.)

Theoremreximi 2650 Inference quantifying both antecedent and consequent. (Contributed by NM, 18-Oct-1996.)

Theoremreximdai 2651 Deduction from Theorem 19.22 of [Margaris] p. 90. (Restricted quantifier version.) (Contributed by NM, 31-Aug-1999.)

Theoremreximdv2 2652* Deduction quantifying both antecedent and consequent, based on Theorem 19.22 of [Margaris] p. 90. (Contributed by NM, 17-Sep-2003.)

Theoremreximdvai 2653* Deduction quantifying both antecedent and consequent, based on Theorem 19.22 of [Margaris] p. 90. (Contributed by NM, 14-Nov-2002.)

Theoremreximdv 2654* Deduction from Theorem 19.22 of [Margaris] p. 90. (Restricted quantifier version with strong hypothesis.) (Contributed by NM, 24-Jun-1998.)

Theoremreximdva 2655* Deduction quantifying both antecedent and consequent, based on Theorem 19.22 of [Margaris] p. 90. (Contributed by NM, 22-May-1999.)

Theoremr19.12 2656* 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 2657 Closed theorem form of r19.23 2658. (Contributed by NM, 4-Mar-2013.) (Revised by Mario Carneiro, 8-Oct-2016.)

Theoremr19.23 2658 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 2659* Theorem 19.23 of [Margaris] p. 90 with restricted quantifiers. (Contributed by NM, 31-Aug-1999.)

Theoremrexlimi 2660 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 2661* Inference from Theorem 19.23 of [Margaris] p. 90. (Restricted quantifier version.) (Contributed by NM, 20-Nov-1994.)

Theoremrexlimiva 2662* Inference from Theorem 19.23 of [Margaris] p. 90 (restricted quantifier version). (Contributed by NM, 18-Dec-2006.)

Theoremrexlimivw 2663* Weaker version of rexlimiv 2661. (Contributed by FL, 19-Sep-2011.)

Theoremrexlimd 2664 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 2665 Version of rexlimd 2664 with deduction version of second hypothesis. (Contributed by NM, 21-Jul-2013.) (Revised by Mario Carneiro, 8-Oct-2016.)

Theoremrexlimdv 2666* 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 2667* Inference from Theorem 19.23 of [Margaris] p. 90 (restricted quantifier version). (Contributed by NM, 20-Jan-2007.)

Theoremrexlimdvaa 2668* Inference from Theorem 19.23 of [Margaris] p. 90 (restricted quantifier version). (Contributed by Mario Carneiro, 15-Jun-2016.)

Theoremrexlimdv3a 2669* Inference from Theorem 19.23 of [Margaris] p. 90 (restricted quantifier version). Frequently-used variant of rexlimdv 2666. (Contributed by NM, 7-Jun-2015.)

Theoremrexlimdvw 2670* Inference from Theorem 19.23 of [Margaris] p. 90 (restricted quantifier version). (Contributed by NM, 18-Jun-2014.)

Theoremrexlimddv 2671* Restricted existential elimination rule of natural deduction. (Contributed by Mario Carneiro, 15-Jun-2016.)

Theoremrexlimivv 2672* Inference from Theorem 19.23 of [Margaris] p. 90 (restricted quantifier version). (Contributed by NM, 17-Feb-2004.)

Theoremrexlimdvv 2673* Inference from Theorem 19.23 of [Margaris] p. 90. (Restricted quantifier version.) (Contributed by NM, 22-Jul-2004.)

Theoremrexlimdvva 2674* Inference from Theorem 19.23 of [Margaris] p. 90. (Restricted quantifier version.) (Contributed by NM, 18-Jun-2014.)

Theoremr19.26 2675 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 2676 Theorem 19.26 of [Margaris] p. 90 with 2 restricted quantifiers. (Contributed by NM, 10-Aug-2004.)

Theoremr19.26-3 2677 Theorem 19.26 of [Margaris] p. 90 with 3 restricted quantifiers. (Contributed by FL, 22-Nov-2010.)

Theoremr19.26m 2678 Theorem 19.26 of [Margaris] p. 90 with mixed quantifiers. (Contributed by NM, 22-Feb-2004.)

Theoremralbi 2679 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 2680 Split a biconditional and distribute quantifier. (Contributed by NM, 3-Jun-2012.)

Theoremr19.27av 2681* 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 2682* 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 2683 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 2684 Variation of Theorem 19.29 of [Margaris] p. 90 with restricted quantifiers. (Contributed by NM, 31-Aug-1999.)

Theoremr19.30 2685 Theorem 19.30 of [Margaris] p. 90 with restricted quantifiers. (Contributed by Scott Fenton, 25-Feb-2011.)

Theoremr19.32v 2686* Theorem 19.32 of [Margaris] p. 90 with restricted quantifiers. (Contributed by NM, 25-Nov-2003.)

Theoremr19.35 2687 Restricted quantifier version of Theorem 19.35 of [Margaris] p. 90. (Contributed by NM, 20-Sep-2003.)

Theoremr19.36av 2688* 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 2689 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.)

Theoremr19.37av 2690* 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 2691 Restricted quantifier version of Theorem 19.40 of [Margaris] p. 90. (Contributed by NM, 2-Apr-2004.)

Theoremr19.41 2692 Restricted quantifier version of Theorem 19.41 of [Margaris] p. 90. (Contributed by NM, 1-Nov-2010.)

Theoremr19.41v 2693* Restricted quantifier version of Theorem 19.41 of [Margaris] p. 90. (Contributed by NM, 17-Dec-2003.)

Theoremr19.42v 2694* Restricted version of Theorem 19.42 of [Margaris] p. 90. (Contributed by NM, 27-May-1998.)

Theoremr19.43 2695 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 2696* 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 2697* 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 2698* Commutation of restricted quantifiers. (Contributed by Mario Carneiro, 14-Oct-2016.)

Theoremrexcomf 2699* Commutation of restricted quantifiers. (Contributed by Mario Carneiro, 14-Oct-2016.)

Theoremralcom 2700* Commutation of restricted quantifiers. (Contributed by NM, 13-Oct-1999.) (Revised by Mario Carneiro, 14-Oct-2016.)

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