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Theorem List for New Foundations Explorer - 2701-2800   *Has distinct variable group(s)
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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.)
(x A (φψ) ↔ (φx A ψ))

Theoremralrimd 2702 Inference from Theorem 19.21 of [Margaris] p. 90. (Restricted quantifier version.) (Contributed by NM, 16-Feb-2004.)
xφ    &   xψ    &   (φ → (ψ → (x Aχ)))       (φ → (ψx A χ))

Theoremralrimdv 2703* Inference from Theorem 19.21 of [Margaris] p. 90. (Restricted quantifier version.) (Contributed by NM, 27-May-1998.)
(φ → (ψ → (x Aχ)))       (φ → (ψx A χ))

Theoremralrimdva 2704* Inference from Theorem 19.21 of [Margaris] p. 90. (Restricted quantifier version.) (Contributed by NM, 2-Feb-2008.)
((φ x A) → (ψχ))       (φ → (ψx A χ))

Theoremralrimivv 2705* Inference from Theorem 19.21 of [Margaris] p. 90. (Restricted quantifier version with double quantification.) (Contributed by NM, 24-Jul-2004.)
(φ → ((x A y B) → ψ))       (φx A y B ψ)

Theoremralrimivva 2706* Inference from Theorem 19.21 of [Margaris] p. 90. (Restricted quantifier version with double quantification.) (Contributed by Jeff Madsen, 19-Jun-2011.)
((φ (x A y B)) → ψ)       (φx A y B ψ)

Theoremralrimivvva 2707* Inference from Theorem 19.21 of [Margaris] p. 90. (Restricted quantifier version with triple quantification.) (Contributed by Mario Carneiro, 9-Jul-2014.)
((φ (x A y B z C)) → ψ)       (φx A y B z C ψ)

Theoremralrimdvv 2708* Inference from Theorem 19.21 of [Margaris] p. 90. (Restricted quantifier version with double quantification.) (Contributed by NM, 1-Jun-2005.)
(φ → (ψ → ((x A y B) → χ)))       (φ → (ψx A y B χ))

Theoremralrimdvva 2709* Inference from Theorem 19.21 of [Margaris] p. 90. (Restricted quantifier version with double quantification.) (Contributed by NM, 2-Feb-2008.)
((φ (x A y B)) → (ψχ))       (φ → (ψx A y B χ))

Theoremrgen2 2710* Generalization rule for restricted quantification. (Contributed by NM, 30-May-1999.)
((x A y B) → φ)       x A y B φ

Theoremrgen3 2711* Generalization rule for restricted quantification. (Contributed by NM, 12-Jan-2008.)
((x A y B z C) → φ)       x A y B z C φ

Theoremr19.21bi 2712 Inference from Theorem 19.21 of [Margaris] p. 90. (Restricted quantifier version.) (Contributed by NM, 20-Nov-1994.)
(φx A ψ)       ((φ x A) → ψ)

Theoremrspec2 2713 Specialization rule for restricted quantification. (Contributed by NM, 20-Nov-1994.)
x A y B φ       ((x A y B) → φ)

Theoremrspec3 2714 Specialization rule for restricted quantification. (Contributed by NM, 20-Nov-1994.)
x A y B z C φ       ((x A y B z C) → φ)

Theoremr19.21be 2715 Inference from Theorem 19.21 of [Margaris] p. 90. (Restricted quantifier version.) (Contributed by NM, 21-Nov-1994.)
(φx A ψ)       x A (φψ)

Theoremnrex 2716 Inference adding restricted existential quantifier to negated wff. (Contributed by NM, 16-Oct-2003.)
(x A → ¬ ψ)        ¬ x A ψ

Theoremnrexdv 2717* Deduction adding restricted existential quantifier to negated wff. (Contributed by NM, 16-Oct-2003.)
((φ x A) → ¬ ψ)       (φ → ¬ x A ψ)

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.)
(x A (φψ) → (x A φx A ψ))

Theoremreximia 2719 Inference quantifying both antecedent and consequent. (Contributed by NM, 10-Feb-1997.)
(x A → (φψ))       (x A φx A ψ)

Theoremreximi2 2720 Inference quantifying both antecedent and consequent, based on Theorem 19.22 of [Margaris] p. 90. (Contributed by NM, 8-Nov-2004.)
((x A φ) → (x B ψ))       (x A φx B ψ)

Theoremreximi 2721 Inference quantifying both antecedent and consequent. (Contributed by NM, 18-Oct-1996.)
(φψ)       (x A φx A ψ)

Theoremreximdai 2722 Deduction from Theorem 19.22 of [Margaris] p. 90. (Restricted quantifier version.) (Contributed by NM, 31-Aug-1999.)
xφ    &   (φ → (x A → (ψχ)))       (φ → (x A ψx A χ))

Theoremreximdv2 2723* Deduction quantifying both antecedent and consequent, based on Theorem 19.22 of [Margaris] p. 90. (Contributed by NM, 17-Sep-2003.)
(φ → ((x A ψ) → (x B χ)))       (φ → (x A ψx B χ))

Theoremreximdvai 2724* Deduction quantifying both antecedent and consequent, based on Theorem 19.22 of [Margaris] p. 90. (Contributed by NM, 14-Nov-2002.)
(φ → (x A → (ψχ)))       (φ → (x A ψx A χ))

Theoremreximdv 2725* Deduction from Theorem 19.22 of [Margaris] p. 90. (Restricted quantifier version with strong hypothesis.) (Contributed by NM, 24-Jun-1998.)
(φ → (ψχ))       (φ → (x A ψx A χ))

Theoremreximdva 2726* Deduction quantifying both antecedent and consequent, based on Theorem 19.22 of [Margaris] p. 90. (Contributed by NM, 22-May-1999.)
((φ x A) → (ψχ))       (φ → (x A ψx A χ))

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.)
(x A y B φy B x A φ)

Theoremr19.23t 2728 Closed theorem form of r19.23 2729. (Contributed by NM, 4-Mar-2013.) (Revised by Mario Carneiro, 8-Oct-2016.)
(Ⅎxψ → (x A (φψ) ↔ (x A φψ)))

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.)
xψ       (x A (φψ) ↔ (x A φψ))

Theoremr19.23v 2730* Theorem 19.23 of [Margaris] p. 90 with restricted quantifiers. (Contributed by NM, 31-Aug-1999.)
(x A (φψ) ↔ (x A φψ))

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.)
xψ    &   (x A → (φψ))       (x A φψ)

Theoremrexlimiv 2732* Inference from Theorem 19.23 of [Margaris] p. 90. (Restricted quantifier version.) (Contributed by NM, 20-Nov-1994.)
(x A → (φψ))       (x A φψ)

Theoremrexlimiva 2733* Inference from Theorem 19.23 of [Margaris] p. 90 (restricted quantifier version). (Contributed by NM, 18-Dec-2006.)
((x A φ) → ψ)       (x A φψ)

Theoremrexlimivw 2734* Weaker version of rexlimiv 2732. (Contributed by FL, 19-Sep-2011.)
(φψ)       (x A φψ)

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.)
xφ    &   xχ    &   (φ → (x A → (ψχ)))       (φ → (x A ψχ))

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.)
xφ    &   (φ → Ⅎxχ)    &   (φ → (x A → (ψχ)))       (φ → (x A ψχ))

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.)
(φ → (x A → (ψχ)))       (φ → (x A ψχ))

Theoremrexlimdva 2738* Inference from Theorem 19.23 of [Margaris] p. 90 (restricted quantifier version). (Contributed by NM, 20-Jan-2007.)
((φ x A) → (ψχ))       (φ → (x A ψχ))

Theoremrexlimdvaa 2739* Inference from Theorem 19.23 of [Margaris] p. 90 (restricted quantifier version). (Contributed by Mario Carneiro, 15-Jun-2016.)
((φ (x A ψ)) → χ)       (φ → (x A ψχ))

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.)
((φ x A ψ) → χ)       (φ → (x A ψχ))

Theoremrexlimdvw 2741* Inference from Theorem 19.23 of [Margaris] p. 90 (restricted quantifier version). (Contributed by NM, 18-Jun-2014.)
(φ → (ψχ))       (φ → (x A ψχ))

Theoremrexlimddv 2742* Restricted existential elimination rule of natural deduction. (Contributed by Mario Carneiro, 15-Jun-2016.)
(φx A ψ)    &   ((φ (x A ψ)) → χ)       (φχ)

Theoremrexlimivv 2743* Inference from Theorem 19.23 of [Margaris] p. 90 (restricted quantifier version). (Contributed by NM, 17-Feb-2004.)
((x A y B) → (φψ))       (x A y B φψ)

Theoremrexlimdvv 2744* Inference from Theorem 19.23 of [Margaris] p. 90. (Restricted quantifier version.) (Contributed by NM, 22-Jul-2004.)
(φ → ((x A y B) → (ψχ)))       (φ → (x A y B ψχ))

Theoremrexlimdvva 2745* Inference from Theorem 19.23 of [Margaris] p. 90. (Restricted quantifier version.) (Contributed by NM, 18-Jun-2014.)
((φ (x A y B)) → (ψχ))       (φ → (x A y B ψχ))

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.)
(x A (φ ψ) ↔ (x A φ x A ψ))

Theoremr19.26-2 2747 Theorem 19.26 of [Margaris] p. 90 with 2 restricted quantifiers. (Contributed by NM, 10-Aug-2004.)
(x A y B (φ ψ) ↔ (x A y B φ x A y B ψ))

Theoremr19.26-3 2748 Theorem 19.26 of [Margaris] p. 90 with 3 restricted quantifiers. (Contributed by FL, 22-Nov-2010.)
(x A (φ ψ χ) ↔ (x A φ x A ψ x A χ))

Theoremr19.26m 2749 Theorem 19.26 of [Margaris] p. 90 with mixed quantifiers. (Contributed by NM, 22-Feb-2004.)
(x((x Aφ) (x Bψ)) ↔ (x A φ x B ψ))

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.)
(x A (φψ) → (x A φx A ψ))

Theoremralbiim 2751 Split a biconditional and distribute quantifier. (Contributed by NM, 3-Jun-2012.)
(x A (φψ) ↔ (x A (φψ) x A (ψφ)))

Theoremr19.27av 2752* Restricted version of one direction of Theorem 19.27 of [Margaris] p. 90. (The other direction doesn't hold when A is empty.) (Contributed by NM, 3-Jun-2004.) (Proof shortened by Andrew Salmon, 30-May-2011.)
((x A φ ψ) → x A (φ ψ))

Theoremr19.28av 2753* Restricted version of one direction of Theorem 19.28 of [Margaris] p. 90. (The other direction doesn't hold when A is empty.) (Contributed by NM, 2-Apr-2004.)
((φ x A ψ) → x A (φ ψ))

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.)
((x A φ x A ψ) → x A (φ ψ))

Theoremr19.29r 2755 Variation of Theorem 19.29 of [Margaris] p. 90 with restricted quantifiers. (Contributed by NM, 31-Aug-1999.)
((x A φ x A ψ) → x A (φ ψ))

Theoremr19.30 2756 Theorem 19.30 of [Margaris] p. 90 with restricted quantifiers. (Contributed by Scott Fenton, 25-Feb-2011.)
(x A (φ ψ) → (x A φ x A ψ))

Theoremr19.32v 2757* Theorem 19.32 of [Margaris] p. 90 with restricted quantifiers. (Contributed by NM, 25-Nov-2003.)
(x A (φ ψ) ↔ (φ x A ψ))

Theoremr19.35 2758 Restricted quantifier version of Theorem 19.35 of [Margaris] p. 90. (Contributed by NM, 20-Sep-2003.)
(x A (φψ) ↔ (x A φx A ψ))

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 A is empty. (Contributed by NM, 22-Oct-2003.)
(x A (φψ) → (x A φψ))

Theoremr19.37 2760 Restricted version of one direction of Theorem 19.37 of [Margaris] p. 90. (The other direction doesn't hold when A is empty.) (Contributed by FL, 13-May-2012.) (Revised by Mario Carneiro, 11-Dec-2016.)
xφ       (x A (φψ) → (φx A ψ))

Theoremr19.37av 2761* Restricted version of one direction of Theorem 19.37 of [Margaris] p. 90. (The other direction doesn't hold when A is empty.) (Contributed by NM, 2-Apr-2004.)
(x A (φψ) → (φx A ψ))

Theoremr19.40 2762 Restricted quantifier version of Theorem 19.40 of [Margaris] p. 90. (Contributed by NM, 2-Apr-2004.)
(x A (φ ψ) → (x A φ x A ψ))

Theoremr19.41 2763 Restricted quantifier version of Theorem 19.41 of [Margaris] p. 90. (Contributed by NM, 1-Nov-2010.)
xψ       (x A (φ ψ) ↔ (x A φ ψ))

Theoremr19.41v 2764* Restricted quantifier version of Theorem 19.41 of [Margaris] p. 90. (Contributed by NM, 17-Dec-2003.)
(x A (φ ψ) ↔ (x A φ ψ))

Theoremr19.42v 2765* Restricted version of Theorem 19.42 of [Margaris] p. 90. (Contributed by NM, 27-May-1998.)
(x A (φ ψ) ↔ (φ x A ψ))

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.)
(x A (φ ψ) ↔ (x A φ x A ψ))

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 A is empty. (Contributed by NM, 2-Apr-2004.)
(x A (φ ψ) → (x A φ ψ))

Theoremr19.45av 2768* Restricted version of one direction of Theorem 19.45 of [Margaris] p. 90. (The other direction doesn't hold when A is empty.) (Contributed by NM, 2-Apr-2004.)
(x A (φ ψ) → (φ x A ψ))

Theoremralcomf 2769* Commutation of restricted quantifiers. (Contributed by Mario Carneiro, 14-Oct-2016.)
yA    &   xB       (x A y B φy B x A φ)

Theoremrexcomf 2770* Commutation of restricted quantifiers. (Contributed by Mario Carneiro, 14-Oct-2016.)
yA    &   xB       (x A y B φy B x A φ)

Theoremralcom 2771* Commutation of restricted quantifiers. (Contributed by NM, 13-Oct-1999.) (Revised by Mario Carneiro, 14-Oct-2016.)
(x A y B φy B x A φ)

Theoremrexcom 2772* Commutation of restricted quantifiers. (Contributed by NM, 19-Nov-1995.) (Revised by Mario Carneiro, 14-Oct-2016.)
(x A y B φy B x A φ)

Theoremrexcom13 2773* Swap 1st and 3rd restricted existential quantifiers. (Contributed by NM, 8-Apr-2015.)
(x A y B z C φz C y B x A φ)

Theoremrexrot4 2774* Rotate existential restricted quantifiers twice. (Contributed by NM, 8-Apr-2015.)
(x A y B z C w D φz C w D x A y B φ)

Theoremralcom2 2775* Commutation of restricted quantifiers. Note that x and y needn't be distinct (this makes the proof longer). (Contributed by NM, 24-Nov-1994.) (Proof shortened by Mario Carneiro, 17-Oct-2016.)
(x A y A φy A x A φ)

Theoremralcom3 2776 A commutative law for restricted quantifiers that swaps the domain of the restriction. (Contributed by NM, 22-Feb-2004.)
(x A (x Bφ) ↔ x B (x Aφ))

Theoremreean 2777* Rearrange existential quantifiers. (Contributed by NM, 27-Oct-2010.) (Proof shortened by Andrew Salmon, 30-May-2011.)
yφ    &   xψ       (x A y B (φ ψ) ↔ (x A φ y B ψ))

Theoremreeanv 2778* Rearrange existential quantifiers. (Contributed by NM, 9-May-1999.)
(x A y B (φ ψ) ↔ (x A φ y B ψ))

Theorem3reeanv 2779* Rearrange three existential quantifiers. (Contributed by Jeff Madsen, 11-Jun-2010.)
(x A y B z C (φ ψ χ) ↔ (x A φ y B ψ z C χ))

Theorem2ralor 2780* Distribute quantification over "or". (Contributed by Jeff Madsen, 19-Jun-2010.)
(x A y B (φ ψ) ↔ (x A φ y B ψ))

Theoremnfreu1 2781 x is not free in ∃!x Aφ. (Contributed by NM, 19-Mar-1997.)
x∃!x A φ

Theoremnfrmo1 2782 x is not free in ∃*x Aφ. (Contributed by NM, 16-Jun-2017.)
x∃*x A φ

Theoremnfreud 2783 Deduction version of nfreu 2785. (Contributed by NM, 15-Feb-2013.) (Revised by Mario Carneiro, 8-Oct-2016.)
yφ    &   (φxA)    &   (φ → Ⅎxψ)       (φ → Ⅎx∃!y A ψ)

Theoremnfrmod 2784 Deduction version of nfrmo 2786. (Contributed by NM, 17-Jun-2017.)
yφ    &   (φxA)    &   (φ → Ⅎxψ)       (φ → Ⅎx∃*y A ψ)

Theoremnfreu 2785 Bound-variable hypothesis builder for restricted uniqueness. (Contributed by NM, 30-Oct-2010.) (Revised by Mario Carneiro, 8-Oct-2016.)
xA    &   xφ       x∃!y A φ

Theoremnfrmo 2786 Bound-variable hypothesis builder for restricted uniqueness. (Contributed by NM, 16-Jun-2017.)
xA    &   xφ       x∃*y A φ

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.)
(x {x A φ} ↔ (x A φ))

Theoremrabid2 2788* An "identity" law for restricted class abstraction. (Contributed by NM, 9-Oct-2003.) (Proof shortened by Andrew Salmon, 30-May-2011.)
(A = {x A φ} ↔ x A φ)

Theoremrabbi 2789 Equivalent wff's correspond to equal restricted class abstractions. Closed theorem form of rabbidva 2850. (Contributed by NM, 25-Nov-2013.)
(x A (ψχ) ↔ {x A ψ} = {x A χ})

Theoremrabswap 2790 Swap with a membership relation in a restricted class abstraction. (Contributed by NM, 4-Jul-2005.)
{x A x B} = {x B x A}

Theoremnfrab1 2791 The abstraction variable in a restricted class abstraction isn't free. (Contributed by NM, 19-Mar-1997.)
x{x A φ}

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.)
xφ    &   xA       x{y A φ}

Theoremreubida 2793 Formula-building rule for restricted existential quantifier (deduction rule). (Contributed by Mario Carneiro, 19-Nov-2016.)
xφ    &   ((φ x A) → (ψχ))       (φ → (∃!x A ψ∃!x A χ))

Theoremreubidva 2794* Formula-building rule for restricted existential quantifier (deduction rule). (Contributed by NM, 13-Nov-2004.)
((φ x A) → (ψχ))       (φ → (∃!x A ψ∃!x A χ))

Theoremreubidv 2795* Formula-building rule for restricted existential quantifier (deduction rule). (Contributed by NM, 17-Oct-1996.)
(φ → (ψχ))       (φ → (∃!x A ψ∃!x A χ))

Theoremreubiia 2796 Formula-building rule for restricted existential quantifier (inference rule). (Contributed by NM, 14-Nov-2004.)
(x A → (φψ))       (∃!x A φ∃!x A ψ)

Theoremreubii 2797 Formula-building rule for restricted existential quantifier (inference rule). (Contributed by NM, 22-Oct-1999.)
(φψ)       (∃!x A φ∃!x A ψ)

Theoremrmobida 2798 Formula-building rule for restricted existential quantifier (deduction rule). (Contributed by NM, 16-Jun-2017.)
xφ    &   ((φ x A) → (ψχ))       (φ → (∃*x A ψ∃*x A χ))

Theoremrmobidva 2799* Formula-building rule for restricted existential quantifier (deduction rule). (Contributed by NM, 16-Jun-2017.)
((φ x A) → (ψχ))       (φ → (∃*x A ψ∃*x A χ))

Theoremrmobidv 2800* Formula-building rule for restricted existential quantifier (deduction rule). (Contributed by NM, 16-Jun-2017.)
(φ → (ψχ))       (φ → (∃*x A ψ∃*x A χ))

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