Theorem List for New Foundations Explorer - 2601-2700 *Has distinct variable
group(s)
Type | Label | Description |
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Theorem | neanior 2601 |
A De Morgan's law for inequality. (Contributed by NM, 18-May-2007.)
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Theorem | ne3anior 2602 |
A De Morgan's law for inequality. (Contributed by NM, 30-Sep-2013.)
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Theorem | neorian 2603 |
A De Morgan's law for inequality. (Contributed by NM, 18-May-2007.)
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Theorem | nemtbir 2604 |
An inference from an inequality, related to modus tollens. (Contributed
by NM, 13-Apr-2007.)
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Theorem | nelne1 2605 |
Two classes are different if they don't contain the same element.
(Contributed by NM, 3-Feb-2012.)
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Theorem | nelne2 2606 |
Two classes are different if they don't belong to the same class.
(Contributed by NM, 25-Jun-2012.)
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Theorem | neleq1 2607 |
Equality theorem for negated membership. (Contributed by NM,
20-Nov-1994.)
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Theorem | neleq2 2608 |
Equality theorem for negated membership. (Contributed by NM,
20-Nov-1994.)
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Theorem | neleq12d 2609 |
Equality theorem for negated membership. (Contributed by FL,
10-Aug-2016.)
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Theorem | nfne 2610 |
Bound-variable hypothesis builder for inequality. (Contributed by NM,
10-Nov-2007.) (Revised by Mario Carneiro, 7-Oct-2016.)
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Theorem | nfnel 2611 |
Bound-variable hypothesis builder for inequality. (Contributed by David
Abernethy, 26-Jun-2011.) (Revised by Mario Carneiro, 7-Oct-2016.)
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Theorem | nfned 2612 |
Bound-variable hypothesis builder for inequality. (Contributed by NM,
10-Nov-2007.) (Revised by Mario Carneiro, 7-Oct-2016.)
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Theorem | nfneld 2613 |
Bound-variable hypothesis builder for inequality. (Contributed by David
Abernethy, 26-Jun-2011.) (Revised by Mario Carneiro, 7-Oct-2016.)
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2.1.5 Restricted quantification
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Syntax | wral 2614 |
Extend wff notation to include restricted universal quantification.
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Syntax | wrex 2615 |
Extend wff notation to include restricted existential quantification.
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Syntax | wreu 2616 |
Extend wff notation to include restricted existential uniqueness.
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Syntax | wrmo 2617 |
Extend wff notation to include restricted "at most one."
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Syntax | crab 2618 |
Extend class notation to include the restricted class abstraction (class
builder).
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Definition | df-ral 2619 |
Define restricted universal quantification. Special case of Definition
4.15(3) of [TakeutiZaring] p. 22.
(Contributed by NM, 19-Aug-1993.)
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Definition | df-rex 2620 |
Define restricted existential quantification. Special case of Definition
4.15(4) of [TakeutiZaring] p. 22.
(Contributed by NM, 30-Aug-1993.)
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Definition | df-reu 2621 |
Define restricted existential uniqueness. (Contributed by NM,
22-Nov-1994.)
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Definition | df-rmo 2622 |
Define restricted "at most one". (Contributed by NM, 16-Jun-2017.)
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Definition | df-rab 2623 |
Define a restricted class abstraction (class builder), which is the class
of all in such that is true. Definition
of
[TakeutiZaring] p. 20. (Contributed
by NM, 22-Nov-1994.)
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Theorem | ralnex 2624 |
Relationship between restricted universal and existential quantifiers.
(Contributed by NM, 21-Jan-1997.)
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Theorem | rexnal 2625 |
Relationship between restricted universal and existential quantifiers.
(Contributed by NM, 21-Jan-1997.)
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Theorem | dfral2 2626 |
Relationship between restricted universal and existential quantifiers.
(Contributed by NM, 21-Jan-1997.)
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Theorem | dfrex2 2627 |
Relationship between restricted universal and existential quantifiers.
(Contributed by NM, 21-Jan-1997.)
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Theorem | ralbida 2628 |
Formula-building rule for restricted universal quantifier (deduction
rule). (Contributed by NM, 6-Oct-2003.)
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Theorem | rexbida 2629 |
Formula-building rule for restricted existential quantifier (deduction
rule). (Contributed by NM, 6-Oct-2003.)
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Theorem | ralbidva 2630* |
Formula-building rule for restricted universal quantifier (deduction
rule). (Contributed by NM, 4-Mar-1997.)
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Theorem | rexbidva 2631* |
Formula-building rule for restricted existential quantifier (deduction
rule). (Contributed by NM, 9-Mar-1997.)
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Theorem | ralbid 2632 |
Formula-building rule for restricted universal quantifier (deduction
rule). (Contributed by NM, 27-Jun-1998.)
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Theorem | rexbid 2633 |
Formula-building rule for restricted existential quantifier (deduction
rule). (Contributed by NM, 27-Jun-1998.)
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Theorem | ralbidv 2634* |
Formula-building rule for restricted universal quantifier (deduction
rule). (Contributed by NM, 20-Nov-1994.)
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Theorem | rexbidv 2635* |
Formula-building rule for restricted existential quantifier (deduction
rule). (Contributed by NM, 20-Nov-1994.)
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Theorem | ralbidv2 2636* |
Formula-building rule for restricted universal quantifier (deduction
rule). (Contributed by NM, 6-Apr-1997.)
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Theorem | rexbidv2 2637* |
Formula-building rule for restricted existential quantifier (deduction
rule). (Contributed by NM, 22-May-1999.)
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Theorem | ralbii 2638 |
Inference adding restricted universal quantifier to both sides of an
equivalence. (Contributed by NM, 23-Nov-1994.) (Revised by Mario
Carneiro, 17-Oct-2016.)
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Theorem | rexbii 2639 |
Inference adding restricted existential quantifier to both sides of an
equivalence. (Contributed by NM, 23-Nov-1994.) (Revised by Mario
Carneiro, 17-Oct-2016.)
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Theorem | 2ralbii 2640 |
Inference adding two restricted universal quantifiers to both sides of
an equivalence. (Contributed by NM, 1-Aug-2004.)
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Theorem | 2rexbii 2641 |
Inference adding two restricted existential quantifiers to both sides of
an equivalence. (Contributed by NM, 11-Nov-1995.)
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Theorem | ralbii2 2642 |
Inference adding different restricted universal quantifiers to each side
of an equivalence. (Contributed by NM, 15-Aug-2005.)
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Theorem | rexbii2 2643 |
Inference adding different restricted existential quantifiers to each
side of an equivalence. (Contributed by NM, 4-Feb-2004.)
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Theorem | raleqbii 2644 |
Equality deduction for restricted universal quantifier, changing both
formula and quantifier domain. Inference form. (Contributed by David
Moews, 1-May-2017.)
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Theorem | rexeqbii 2645 |
Equality deduction for restricted existential quantifier, changing both
formula and quantifier domain. Inference form. (Contributed by David
Moews, 1-May-2017.)
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Theorem | ralbiia 2646 |
Inference adding restricted universal quantifier to both sides of an
equivalence. (Contributed by NM, 26-Nov-2000.)
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Theorem | rexbiia 2647 |
Inference adding restricted existential quantifier to both sides of an
equivalence. (Contributed by NM, 26-Oct-1999.)
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Theorem | 2rexbiia 2648* |
Inference adding two restricted existential quantifiers to both sides of
an equivalence. (Contributed by NM, 1-Aug-2004.)
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Theorem | r2alf 2649* |
Double restricted universal quantification. (Contributed by Mario
Carneiro, 14-Oct-2016.)
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Theorem | r2exf 2650* |
Double restricted existential quantification. (Contributed by Mario
Carneiro, 14-Oct-2016.)
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Theorem | r2al 2651* |
Double restricted universal quantification. (Contributed by NM,
19-Nov-1995.)
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Theorem | r2ex 2652* |
Double restricted existential quantification. (Contributed by NM,
11-Nov-1995.)
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Theorem | 2ralbida 2653* |
Formula-building rule for restricted universal quantifier (deduction
rule). (Contributed by NM, 24-Feb-2004.)
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Theorem | 2ralbidva 2654* |
Formula-building rule for restricted universal quantifiers (deduction
rule). (Contributed by NM, 4-Mar-1997.)
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Theorem | 2rexbidva 2655* |
Formula-building rule for restricted existential quantifiers (deduction
rule). (Contributed by NM, 15-Dec-2004.)
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Theorem | 2ralbidv 2656* |
Formula-building rule for restricted universal quantifiers (deduction
rule). (Contributed by NM, 28-Jan-2006.) (Revised by Szymon
Jaroszewicz, 16-Mar-2007.)
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Theorem | 2rexbidv 2657* |
Formula-building rule for restricted existential quantifiers (deduction
rule). (Contributed by NM, 28-Jan-2006.)
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Theorem | rexralbidv 2658* |
Formula-building rule for restricted quantifiers (deduction rule).
(Contributed by NM, 28-Jan-2006.)
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Theorem | ralinexa 2659 |
A transformation of restricted quantifiers and logical connectives.
(Contributed by NM, 4-Sep-2005.)
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Theorem | rexanali 2660 |
A transformation of restricted quantifiers and logical connectives.
(Contributed by NM, 4-Sep-2005.)
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Theorem | risset 2661* |
Two ways to say " belongs to ." (Contributed by NM,
22-Nov-1994.)
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Theorem | hbral 2662 |
Bound-variable hypothesis builder for restricted quantification.
(Contributed by NM, 1-Sep-1999.) (Revised by David Abernethy,
13-Dec-2009.)
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Theorem | hbra1 2663 |
is not free in .
(Contributed by NM,
18-Oct-1996.)
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Theorem | nfra1 2664 |
is not free in .
(Contributed by NM, 18-Oct-1996.)
(Revised by Mario Carneiro, 7-Oct-2016.)
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Theorem | nfrald 2665 |
Deduction version of nfral 2667. (Contributed by NM, 15-Feb-2013.)
(Revised by Mario Carneiro, 7-Oct-2016.)
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Theorem | nfrexd 2666 |
Deduction version of nfrex 2669. (Contributed by Mario Carneiro,
14-Oct-2016.)
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Theorem | nfral 2667 |
Bound-variable hypothesis builder for restricted quantification.
(Contributed by NM, 1-Sep-1999.) (Revised by Mario Carneiro,
7-Oct-2016.)
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Theorem | nfra2 2668* |
Similar to Lemma 24 of [Monk2] p. 114, except the
quantification of the
antecedent is restricted. Derived automatically from hbra2VD in set.mm.
Contributed by Alan Sare 31-Dec-2011. (Contributed by NM,
31-Dec-2011.)
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Theorem | nfrex 2669 |
Bound-variable hypothesis builder for restricted quantification.
(Contributed by NM, 1-Sep-1999.) (Revised by Mario Carneiro,
7-Oct-2016.)
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Theorem | nfre1 2670 |
is not free in .
(Contributed by NM, 19-Mar-1997.)
(Revised by Mario Carneiro, 7-Oct-2016.)
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Theorem | r3al 2671* |
Triple restricted universal quantification. (Contributed by NM,
19-Nov-1995.)
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Theorem | alral 2672 |
Universal quantification implies restricted quantification. (Contributed
by NM, 20-Oct-2006.)
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Theorem | rexex 2673 |
Restricted existence implies existence. (Contributed by NM,
11-Nov-1995.)
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Theorem | rsp 2674 |
Restricted specialization. (Contributed by NM, 17-Oct-1996.)
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Theorem | rspe 2675 |
Restricted specialization. (Contributed by NM, 12-Oct-1999.)
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Theorem | rsp2 2676 |
Restricted specialization. (Contributed by NM, 11-Feb-1997.)
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Theorem | rsp2e 2677 |
Restricted specialization. (Contributed by FL, 4-Jun-2012.)
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Theorem | rspec 2678 |
Specialization rule for restricted quantification. (Contributed by NM,
19-Nov-1994.)
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Theorem | rgen 2679 |
Generalization rule for restricted quantification. (Contributed by NM,
19-Nov-1994.)
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Theorem | rgen2a 2680* |
Generalization rule for restricted quantification. Note that and
needn't be
distinct (and illustrates the use of dvelim 2016).
(Contributed by NM, 23-Nov-1994.) (Proof shortened by Andrew Salmon,
25-May-2011.) (Proof modification is discouraged.
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Theorem | rgenw 2681 |
Generalization rule for restricted quantification. (Contributed by NM,
18-Jun-2014.)
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Theorem | rgen2w 2682 |
Generalization rule for restricted quantification. Note that and
needn't be
distinct. (Contributed by NM, 18-Jun-2014.)
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Theorem | mprg 2683 |
Modus ponens combined with restricted generalization. (Contributed by
NM, 10-Aug-2004.)
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Theorem | mprgbir 2684 |
Modus ponens on biconditional combined with restricted generalization.
(Contributed by NM, 21-Mar-2004.)
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Theorem | ralim 2685 |
Distribution of restricted quantification over implication. (Contributed
by NM, 9-Feb-1997.)
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Theorem | ralimi2 2686 |
Inference quantifying both antecedent and consequent. (Contributed by
NM, 22-Feb-2004.)
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Theorem | ralimia 2687 |
Inference quantifying both antecedent and consequent. (Contributed by
NM, 19-Jul-1996.)
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Theorem | ralimiaa 2688 |
Inference quantifying both antecedent and consequent. (Contributed by
NM, 4-Aug-2007.)
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Theorem | ralimi 2689 |
Inference quantifying both antecedent and consequent, with strong
hypothesis. (Contributed by NM, 4-Mar-1997.)
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Theorem | ral2imi 2690 |
Inference quantifying antecedent, nested antecedent, and consequent,
with a strong hypothesis. (Contributed by NM, 19-Dec-2006.)
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Theorem | ralimdaa 2691 |
Deduction quantifying both antecedent and consequent, based on Theorem
19.20 of [Margaris] p. 90.
(Contributed by NM, 22-Sep-2003.)
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Theorem | ralimdva 2692* |
Deduction quantifying both antecedent and consequent, based on Theorem
19.20 of [Margaris] p. 90.
(Contributed by NM, 22-May-1999.)
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Theorem | ralimdv 2693* |
Deduction quantifying both antecedent and consequent, based on Theorem
19.20 of [Margaris] p. 90.
(Contributed by NM, 8-Oct-2003.)
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Theorem | ralimdv2 2694* |
Inference quantifying both antecedent and consequent. (Contributed by
NM, 1-Feb-2005.)
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Theorem | ralrimi 2695 |
Inference from Theorem 19.21 of [Margaris] p.
90 (restricted quantifier
version). (Contributed by NM, 10-Oct-1999.)
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Theorem | ralrimiv 2696* |
Inference from Theorem 19.21 of [Margaris] p.
90. (Restricted
quantifier version.) (Contributed by NM, 22-Nov-1994.)
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Theorem | ralrimiva 2697* |
Inference from Theorem 19.21 of [Margaris] p.
90. (Restricted
quantifier version.) (Contributed by NM, 2-Jan-2006.)
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Theorem | ralrimivw 2698* |
Inference from Theorem 19.21 of [Margaris] p.
90. (Restricted
quantifier version.) (Contributed by NM, 18-Jun-2014.)
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Theorem | r19.21t 2699 |
Theorem 19.21 of [Margaris] p. 90 with
restricted quantifiers (closed
theorem version). (Contributed by NM, 1-Mar-2008.)
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Theorem | r19.21 2700 |
Theorem 19.21 of [Margaris] p. 90 with
restricted quantifiers.
(Contributed by Scott Fenton, 30-Mar-2011.)
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