Theorem List for Intuitionistic Logic Explorer - 2501-2600 *Has distinct variable
group(s)
Type | Label | Description |
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Theorem | ralrimi 2501 |
Inference from Theorem 19.21 of [Margaris] p.
90 (restricted quantifier
version). (Contributed by NM, 10-Oct-1999.)
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Theorem | ralrimiv 2502* |
Inference from Theorem 19.21 of [Margaris] p.
90. (Restricted
quantifier version.) (Contributed by NM, 22-Nov-1994.)
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Theorem | ralrimiva 2503* |
Inference from Theorem 19.21 of [Margaris] p.
90. (Restricted
quantifier version.) (Contributed by NM, 2-Jan-2006.)
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Theorem | ralrimivw 2504* |
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 2505 |
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 2506 |
Theorem 19.21 of [Margaris] p. 90 with
restricted quantifiers.
(Contributed by Scott Fenton, 30-Mar-2011.)
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Theorem | r19.21v 2507* |
Theorem 19.21 of [Margaris] p. 90 with
restricted quantifiers.
(Contributed by NM, 15-Oct-2003.) (Proof shortened by Andrew Salmon,
30-May-2011.)
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Theorem | ralrimd 2508 |
Inference from Theorem 19.21 of [Margaris] p.
90. (Restricted
quantifier version.) (Contributed by NM, 16-Feb-2004.)
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Theorem | ralrimdv 2509* |
Inference from Theorem 19.21 of [Margaris] p.
90. (Restricted
quantifier version.) (Contributed by NM, 27-May-1998.)
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Theorem | ralrimdva 2510* |
Inference from Theorem 19.21 of [Margaris] p.
90. (Restricted
quantifier version.) (Contributed by NM, 2-Feb-2008.)
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Theorem | ralrimivv 2511* |
Inference from Theorem 19.21 of [Margaris] p.
90. (Restricted
quantifier version with double quantification.) (Contributed by NM,
24-Jul-2004.)
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Theorem | ralrimivva 2512* |
Inference from Theorem 19.21 of [Margaris] p.
90. (Restricted
quantifier version with double quantification.) (Contributed by Jeff
Madsen, 19-Jun-2011.)
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Theorem | ralrimivvva 2513* |
Inference from Theorem 19.21 of [Margaris] p.
90. (Restricted
quantifier version with triple quantification.) (Contributed by Mario
Carneiro, 9-Jul-2014.)
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Theorem | ralrimdvv 2514* |
Inference from Theorem 19.21 of [Margaris] p.
90. (Restricted
quantifier version with double quantification.) (Contributed by NM,
1-Jun-2005.)
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Theorem | ralrimdvva 2515* |
Inference from Theorem 19.21 of [Margaris] p.
90. (Restricted
quantifier version with double quantification.) (Contributed by NM,
2-Feb-2008.)
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Theorem | rgen2 2516* |
Generalization rule for restricted quantification. (Contributed by NM,
30-May-1999.)
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Theorem | rgen3 2517* |
Generalization rule for restricted quantification. (Contributed by NM,
12-Jan-2008.)
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Theorem | r19.21bi 2518 |
Inference from Theorem 19.21 of [Margaris] p.
90. (Restricted
quantifier version.) (Contributed by NM, 20-Nov-1994.)
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Theorem | rspec2 2519 |
Specialization rule for restricted quantification. (Contributed by NM,
20-Nov-1994.)
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Theorem | rspec3 2520 |
Specialization rule for restricted quantification. (Contributed by NM,
20-Nov-1994.)
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Theorem | r19.21be 2521 |
Inference from Theorem 19.21 of [Margaris] p.
90. (Restricted
quantifier version.) (Contributed by NM, 21-Nov-1994.)
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Theorem | nrex 2522 |
Inference adding restricted existential quantifier to negated wff.
(Contributed by NM, 16-Oct-2003.)
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Theorem | nrexdv 2523* |
Deduction adding restricted existential quantifier to negated wff.
(Contributed by NM, 16-Oct-2003.)
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Theorem | rexim 2524 |
Theorem 19.22 of [Margaris] p. 90.
(Restricted quantifier version.)
(Contributed by NM, 22-Nov-1994.) (Proof shortened by Andrew Salmon,
30-May-2011.)
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Theorem | reximia 2525 |
Inference quantifying both antecedent and consequent. (Contributed by
NM, 10-Feb-1997.)
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Theorem | reximi2 2526 |
Inference quantifying both antecedent and consequent, based on Theorem
19.22 of [Margaris] p. 90.
(Contributed by NM, 8-Nov-2004.)
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Theorem | reximi 2527 |
Inference quantifying both antecedent and consequent. (Contributed by
NM, 18-Oct-1996.)
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Theorem | reximdai 2528 |
Deduction from Theorem 19.22 of [Margaris] p.
90. (Restricted
quantifier version.) (Contributed by NM, 31-Aug-1999.)
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Theorem | reximdv2 2529* |
Deduction quantifying both antecedent and consequent, based on Theorem
19.22 of [Margaris] p. 90.
(Contributed by NM, 17-Sep-2003.)
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Theorem | reximdvai 2530* |
Deduction quantifying both antecedent and consequent, based on Theorem
19.22 of [Margaris] p. 90.
(Contributed by NM, 14-Nov-2002.)
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Theorem | reximdv 2531* |
Deduction from Theorem 19.22 of [Margaris] p.
90. (Restricted
quantifier version with strong hypothesis.) (Contributed by NM,
24-Jun-1998.)
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Theorem | reximdva 2532* |
Deduction quantifying both antecedent and consequent, based on Theorem
19.22 of [Margaris] p. 90.
(Contributed by NM, 22-May-1999.)
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Theorem | reximddv 2533* |
Deduction from Theorem 19.22 of [Margaris] p.
90. (Contributed by
Thierry Arnoux, 7-Dec-2016.)
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Theorem | reximssdv 2534* |
Derivation of a restricted existential quantification over a subset (the
second hypothesis implies
), deduction form.
(Contributed by
AV, 21-Aug-2022.)
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Theorem | reximddv2 2535* |
Double deduction from Theorem 19.22 of [Margaris] p. 90. (Contributed
by Thierry Arnoux, 15-Dec-2019.)
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Theorem | r19.12 2536* |
Theorem 19.12 of [Margaris] p. 89 with
restricted quantifiers.
(Contributed by NM, 15-Oct-2003.) (Proof shortened by Andrew Salmon,
30-May-2011.)
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Theorem | r19.23t 2537 |
Closed theorem form of r19.23 2538. (Contributed by NM, 4-Mar-2013.)
(Revised by Mario Carneiro, 8-Oct-2016.)
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Theorem | r19.23 2538 |
Theorem 19.23 of [Margaris] p. 90 with
restricted quantifiers.
(Contributed by NM, 22-Oct-2010.) (Proof shortened by Mario Carneiro,
8-Oct-2016.)
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Theorem | r19.23v 2539* |
Theorem 19.23 of [Margaris] p. 90 with
restricted quantifiers.
(Contributed by NM, 31-Aug-1999.)
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Theorem | rexlimi 2540 |
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.)
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Theorem | rexlimiv 2541* |
Inference from Theorem 19.23 of [Margaris] p.
90. (Restricted
quantifier version.) (Contributed by NM, 20-Nov-1994.)
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Theorem | rexlimiva 2542* |
Inference from Theorem 19.23 of [Margaris] p.
90 (restricted quantifier
version). (Contributed by NM, 18-Dec-2006.)
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Theorem | rexlimivw 2543* |
Weaker version of rexlimiv 2541. (Contributed by FL, 19-Sep-2011.)
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Theorem | rexlimd 2544 |
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.)
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Theorem | rexlimd2 2545 |
Version of rexlimd 2544 with deduction version of second hypothesis.
(Contributed by NM, 21-Jul-2013.) (Revised by Mario Carneiro,
8-Oct-2016.)
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Theorem | rexlimdv 2546* |
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.)
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Theorem | rexlimdva 2547* |
Inference from Theorem 19.23 of [Margaris] p.
90 (restricted quantifier
version). (Contributed by NM, 20-Jan-2007.)
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Theorem | rexlimdvaa 2548* |
Inference from Theorem 19.23 of [Margaris] p.
90 (restricted quantifier
version). (Contributed by Mario Carneiro, 15-Jun-2016.)
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Theorem | rexlimdv3a 2549* |
Inference from Theorem 19.23 of [Margaris] p.
90 (restricted quantifier
version). Frequently-used variant of rexlimdv 2546. (Contributed by NM,
7-Jun-2015.)
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Theorem | rexlimdva2 2550* |
Inference from Theorem 19.23 of [Margaris] p.
90 (restricted quantifier
version). (Contributed by Glauco Siliprandi, 2-Jan-2022.)
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Theorem | rexlimdvw 2551* |
Inference from Theorem 19.23 of [Margaris] p.
90 (restricted quantifier
version). (Contributed by NM, 18-Jun-2014.)
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Theorem | rexlimddv 2552* |
Restricted existential elimination rule of natural deduction.
(Contributed by Mario Carneiro, 15-Jun-2016.)
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Theorem | rexlimivv 2553* |
Inference from Theorem 19.23 of [Margaris] p.
90 (restricted quantifier
version). (Contributed by NM, 17-Feb-2004.)
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Theorem | rexlimdvv 2554* |
Inference from Theorem 19.23 of [Margaris] p.
90. (Restricted
quantifier version.) (Contributed by NM, 22-Jul-2004.)
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Theorem | rexlimdvva 2555* |
Inference from Theorem 19.23 of [Margaris] p.
90. (Restricted
quantifier version.) (Contributed by NM, 18-Jun-2014.)
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Theorem | r19.26 2556 |
Theorem 19.26 of [Margaris] p. 90 with
restricted quantifiers.
(Contributed by NM, 28-Jan-1997.) (Proof shortened by Andrew Salmon,
30-May-2011.)
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Theorem | r19.27v 2557* |
Restricted quantitifer version of one direction of 19.27 1540. (The other
direction holds when is inhabited, see r19.27mv 3454.) (Contributed
by NM, 3-Jun-2004.) (Proof shortened by Andrew Salmon, 30-May-2011.)
(Proof shortened by Wolf Lammen, 17-Jun-2023.)
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Theorem | r19.28v 2558* |
Restricted quantifier version of one direction of 19.28 1542. (The other
direction holds when is inhabited, see r19.28mv 3450.) (Contributed
by NM, 2-Apr-2004.) (Proof shortened by Wolf Lammen, 17-Jun-2023.)
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Theorem | r19.26-2 2559 |
Theorem 19.26 of [Margaris] p. 90 with 2
restricted quantifiers.
(Contributed by NM, 10-Aug-2004.)
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Theorem | r19.26-3 2560 |
Theorem 19.26 of [Margaris] p. 90 with 3
restricted quantifiers.
(Contributed by FL, 22-Nov-2010.)
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Theorem | r19.26m 2561 |
Theorem 19.26 of [Margaris] p. 90 with mixed
quantifiers. (Contributed by
NM, 22-Feb-2004.)
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Theorem | ralbi 2562 |
Distribute a restricted universal quantifier over a biconditional.
Theorem 19.15 of [Margaris] p. 90 with
restricted quantification.
(Contributed by NM, 6-Oct-2003.)
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Theorem | rexbi 2563 |
Distribute a restricted existential quantifier over a biconditional.
Theorem 19.18 of [Margaris] p. 90 with
restricted quantification.
(Contributed by Jim Kingdon, 21-Jan-2019.)
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Theorem | ralbiim 2564 |
Split a biconditional and distribute quantifier. (Contributed by NM,
3-Jun-2012.)
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Theorem | r19.27av 2565* |
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.)
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Theorem | r19.28av 2566* |
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.)
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Theorem | r19.29 2567 |
Theorem 19.29 of [Margaris] p. 90 with
restricted quantifiers.
(Contributed by NM, 31-Aug-1999.) (Proof shortened by Andrew Salmon,
30-May-2011.)
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Theorem | r19.29r 2568 |
Variation of Theorem 19.29 of [Margaris] p. 90
with restricted
quantifiers. (Contributed by NM, 31-Aug-1999.)
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Theorem | ralnex2 2569 |
Relationship between two restricted universal and existential quantifiers.
(Contributed by Glauco Siliprandi, 11-Dec-2019.) (Proof shortened by Wolf
Lammen, 18-May-2023.)
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Theorem | r19.29af2 2570 |
A commonly used pattern based on r19.29 2567 (Contributed by Thierry
Arnoux, 17-Dec-2017.)
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Theorem | r19.29af 2571* |
A commonly used pattern based on r19.29 2567 (Contributed by Thierry
Arnoux, 29-Nov-2017.)
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Theorem | r19.29an 2572* |
A commonly used pattern based on r19.29 2567. (Contributed by Thierry
Arnoux, 29-Dec-2019.)
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Theorem | r19.29a 2573* |
A commonly used pattern based on r19.29 2567 (Contributed by Thierry
Arnoux, 22-Nov-2017.)
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Theorem | r19.29d2r 2574 |
Theorem 19.29 of [Margaris] p. 90 with two
restricted quantifiers,
deduction version (Contributed by Thierry Arnoux, 30-Jan-2017.)
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Theorem | r19.29vva 2575* |
A commonly used pattern based on r19.29 2567, version with two restricted
quantifiers. (Contributed by Thierry Arnoux, 26-Nov-2017.)
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Theorem | r19.32r 2576 |
One direction of Theorem 19.32 of [Margaris]
p. 90 with restricted
quantifiers. For decidable propositions this is an equivalence.
(Contributed by Jim Kingdon, 19-Aug-2018.)
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Theorem | r19.32vr 2577* |
One direction of Theorem 19.32 of [Margaris]
p. 90 with restricted
quantifiers. For decidable propositions this is an equivalence, as seen
at r19.32vdc 2578. (Contributed by Jim Kingdon, 19-Aug-2018.)
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Theorem | r19.32vdc 2578* |
Theorem 19.32 of [Margaris] p. 90 with
restricted quantifiers, where
is
decidable. (Contributed by Jim Kingdon, 4-Jun-2018.)
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DECID |
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Theorem | r19.35-1 2579 |
Restricted quantifier version of 19.35-1 1603. (Contributed by Jim Kingdon,
4-Jun-2018.)
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Theorem | r19.36av 2580* |
One direction of a restricted quantifier version of Theorem 19.36 of
[Margaris] p. 90. In classical logic,
the converse would hold if
has at least one element, but in intuitionistic logic, that is not a
sufficient condition. (Contributed by NM, 22-Oct-2003.)
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Theorem | r19.37 2581 |
Restricted version of one direction of Theorem 19.37 of [Margaris]
p. 90. In classical logic the converse would hold if has at least
one element, but that is not sufficient in intuitionistic logic.
(Contributed by FL, 13-May-2012.) (Revised by Mario Carneiro,
11-Dec-2016.)
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Theorem | r19.37av 2582* |
Restricted version of one direction of Theorem 19.37 of [Margaris]
p. 90. (Contributed by NM, 2-Apr-2004.)
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Theorem | r19.40 2583 |
Restricted quantifier version of Theorem 19.40 of [Margaris] p. 90.
(Contributed by NM, 2-Apr-2004.)
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Theorem | r19.41 2584 |
Restricted quantifier version of Theorem 19.41 of [Margaris] p. 90.
(Contributed by NM, 1-Nov-2010.)
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Theorem | r19.41v 2585* |
Restricted quantifier version of Theorem 19.41 of [Margaris] p. 90.
(Contributed by NM, 17-Dec-2003.)
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Theorem | r19.42v 2586* |
Restricted version of Theorem 19.42 of [Margaris] p. 90. (Contributed
by NM, 27-May-1998.)
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Theorem | r19.43 2587 |
Restricted version of Theorem 19.43 of [Margaris] p. 90. (Contributed by
NM, 27-May-1998.) (Proof rewritten by Jim Kingdon, 5-Jun-2018.)
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Theorem | r19.44av 2588* |
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.)
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Theorem | r19.45av 2589* |
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.)
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Theorem | ralcomf 2590* |
Commutation of restricted quantifiers. (Contributed by Mario Carneiro,
14-Oct-2016.)
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Theorem | rexcomf 2591* |
Commutation of restricted quantifiers. (Contributed by Mario Carneiro,
14-Oct-2016.)
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Theorem | ralcom 2592* |
Commutation of restricted quantifiers. (Contributed by NM,
13-Oct-1999.) (Revised by Mario Carneiro, 14-Oct-2016.)
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Theorem | rexcom 2593* |
Commutation of restricted quantifiers. (Contributed by NM,
19-Nov-1995.) (Revised by Mario Carneiro, 14-Oct-2016.)
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Theorem | rexcom13 2594* |
Swap 1st and 3rd restricted existential quantifiers. (Contributed by
NM, 8-Apr-2015.)
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Theorem | rexrot4 2595* |
Rotate existential restricted quantifiers twice. (Contributed by NM,
8-Apr-2015.)
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Theorem | ralcom3 2596 |
A commutative law for restricted quantifiers that swaps the domain of the
restriction. (Contributed by NM, 22-Feb-2004.)
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Theorem | reean 2597* |
Rearrange existential quantifiers. (Contributed by NM, 27-Oct-2010.)
(Proof shortened by Andrew Salmon, 30-May-2011.)
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Theorem | reeanv 2598* |
Rearrange existential quantifiers. (Contributed by NM, 9-May-1999.)
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Theorem | 3reeanv 2599* |
Rearrange three existential quantifiers. (Contributed by Jeff Madsen,
11-Jun-2010.)
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Theorem | nfreu1 2600 |
is not free in .
(Contributed by NM,
19-Mar-1997.)
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