Theorem List for Intuitionistic Logic Explorer - 2401-2500 *Has distinct variable
group(s)
Type | Label | Description |
Statement |
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Theorem | nfrexdya 2401* |
Not-free for restricted existential quantification where and
are distinct. See nfrexdxy 2399 for a version with and
distinct instead. (Contributed by Jim Kingdon, 30-May-2018.)
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Theorem | nfralxy 2402* |
Not-free for restricted universal quantification where and
are distinct. See nfralya 2404 for a version with and distinct
instead. (Contributed by Jim Kingdon, 30-May-2018.)
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Theorem | nfrexxy 2403* |
Not-free for restricted existential quantification where and
are distinct. See nfrexya 2405 for a version with and distinct
instead. (Contributed by Jim Kingdon, 30-May-2018.)
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Theorem | nfralya 2404* |
Not-free for restricted universal quantification where and
are distinct. See nfralxy 2402 for a version with and distinct
instead. (Contributed by Jim Kingdon, 3-Jun-2018.)
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Theorem | nfrexya 2405* |
Not-free for restricted existential quantification where and
are distinct. See nfrexxy 2403 for a version with and distinct
instead. (Contributed by Jim Kingdon, 3-Jun-2018.)
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Theorem | nfra2xy 2406* |
Not-free given two restricted quantifiers. (Contributed by Jim Kingdon,
20-Aug-2018.)
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Theorem | nfre1 2407 |
is not free in .
(Contributed by NM, 19-Mar-1997.)
(Revised by Mario Carneiro, 7-Oct-2016.)
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Theorem | r3al 2408* |
Triple restricted universal quantification. (Contributed by NM,
19-Nov-1995.)
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Theorem | alral 2409 |
Universal quantification implies restricted quantification. (Contributed
by NM, 20-Oct-2006.)
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Theorem | rexex 2410 |
Restricted existence implies existence. (Contributed by NM,
11-Nov-1995.)
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Theorem | rsp 2411 |
Restricted specialization. (Contributed by NM, 17-Oct-1996.)
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Theorem | rspe 2412 |
Restricted specialization. (Contributed by NM, 12-Oct-1999.)
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Theorem | rsp2 2413 |
Restricted specialization. (Contributed by NM, 11-Feb-1997.)
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Theorem | rsp2e 2414 |
Restricted specialization. (Contributed by FL, 4-Jun-2012.)
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Theorem | rspec 2415 |
Specialization rule for restricted quantification. (Contributed by NM,
19-Nov-1994.)
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Theorem | rgen 2416 |
Generalization rule for restricted quantification. (Contributed by NM,
19-Nov-1994.)
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Theorem | rgen2a 2417* |
Generalization rule for restricted quantification. Note that and
needn't be
distinct (and illustrates the use of dvelimor 1935).
(Contributed by NM, 23-Nov-1994.) (Proof rewritten by Jim Kingdon,
1-Jun-2018.)
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Theorem | rgenw 2418 |
Generalization rule for restricted quantification. (Contributed by NM,
18-Jun-2014.)
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Theorem | rgen2w 2419 |
Generalization rule for restricted quantification. Note that and
needn't be
distinct. (Contributed by NM, 18-Jun-2014.)
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Theorem | mprg 2420 |
Modus ponens combined with restricted generalization. (Contributed by
NM, 10-Aug-2004.)
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Theorem | mprgbir 2421 |
Modus ponens on biconditional combined with restricted generalization.
(Contributed by NM, 21-Mar-2004.)
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Theorem | ralim 2422 |
Distribution of restricted quantification over implication. (Contributed
by NM, 9-Feb-1997.)
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Theorem | ralimi2 2423 |
Inference quantifying both antecedent and consequent. (Contributed by
NM, 22-Feb-2004.)
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Theorem | ralimia 2424 |
Inference quantifying both antecedent and consequent. (Contributed by
NM, 19-Jul-1996.)
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Theorem | ralimiaa 2425 |
Inference quantifying both antecedent and consequent. (Contributed by
NM, 4-Aug-2007.)
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Theorem | ralimi 2426 |
Inference quantifying both antecedent and consequent, with strong
hypothesis. (Contributed by NM, 4-Mar-1997.)
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Theorem | ral2imi 2427 |
Inference quantifying antecedent, nested antecedent, and consequent,
with a strong hypothesis. (Contributed by NM, 19-Dec-2006.)
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Theorem | ralimdaa 2428 |
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 2429* |
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 2430* |
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 2431* |
Inference quantifying both antecedent and consequent. (Contributed by
NM, 1-Feb-2005.)
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Theorem | ralrimi 2432 |
Inference from Theorem 19.21 of [Margaris] p.
90 (restricted quantifier
version). (Contributed by NM, 10-Oct-1999.)
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Theorem | ralrimiv 2433* |
Inference from Theorem 19.21 of [Margaris] p.
90. (Restricted
quantifier version.) (Contributed by NM, 22-Nov-1994.)
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Theorem | ralrimiva 2434* |
Inference from Theorem 19.21 of [Margaris] p.
90. (Restricted
quantifier version.) (Contributed by NM, 2-Jan-2006.)
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Theorem | ralrimivw 2435* |
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 2436 |
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 2437 |
Theorem 19.21 of [Margaris] p. 90 with
restricted quantifiers.
(Contributed by Scott Fenton, 30-Mar-2011.)
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Theorem | r19.21v 2438* |
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 2439 |
Inference from Theorem 19.21 of [Margaris] p.
90. (Restricted
quantifier version.) (Contributed by NM, 16-Feb-2004.)
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Theorem | ralrimdv 2440* |
Inference from Theorem 19.21 of [Margaris] p.
90. (Restricted
quantifier version.) (Contributed by NM, 27-May-1998.)
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Theorem | ralrimdva 2441* |
Inference from Theorem 19.21 of [Margaris] p.
90. (Restricted
quantifier version.) (Contributed by NM, 2-Feb-2008.)
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Theorem | ralrimivv 2442* |
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 2443* |
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 2444* |
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 2445* |
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 2446* |
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 2447* |
Generalization rule for restricted quantification. (Contributed by NM,
30-May-1999.)
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Theorem | rgen3 2448* |
Generalization rule for restricted quantification. (Contributed by NM,
12-Jan-2008.)
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Theorem | r19.21bi 2449 |
Inference from Theorem 19.21 of [Margaris] p.
90. (Restricted
quantifier version.) (Contributed by NM, 20-Nov-1994.)
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Theorem | rspec2 2450 |
Specialization rule for restricted quantification. (Contributed by NM,
20-Nov-1994.)
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Theorem | rspec3 2451 |
Specialization rule for restricted quantification. (Contributed by NM,
20-Nov-1994.)
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Theorem | r19.21be 2452 |
Inference from Theorem 19.21 of [Margaris] p.
90. (Restricted
quantifier version.) (Contributed by NM, 21-Nov-1994.)
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Theorem | nrex 2453 |
Inference adding restricted existential quantifier to negated wff.
(Contributed by NM, 16-Oct-2003.)
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Theorem | nrexdv 2454* |
Deduction adding restricted existential quantifier to negated wff.
(Contributed by NM, 16-Oct-2003.)
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Theorem | rexim 2455 |
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 2456 |
Inference quantifying both antecedent and consequent. (Contributed by
NM, 10-Feb-1997.)
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Theorem | reximi2 2457 |
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 2458 |
Inference quantifying both antecedent and consequent. (Contributed by
NM, 18-Oct-1996.)
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Theorem | reximdai 2459 |
Deduction from Theorem 19.22 of [Margaris] p.
90. (Restricted
quantifier version.) (Contributed by NM, 31-Aug-1999.)
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Theorem | reximdv2 2460* |
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 2461* |
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 2462* |
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 2463* |
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 2464* |
Deduction from Theorem 19.22 of [Margaris] p.
90. (Contributed by
Thierry Arnoux, 7-Dec-2016.)
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Theorem | reximddv2 2465* |
Double deduction from Theorem 19.22 of [Margaris] p. 90. (Contributed
by Thierry Arnoux, 15-Dec-2019.)
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Theorem | r19.12 2466* |
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 2467 |
Closed theorem form of r19.23 2468. (Contributed by NM, 4-Mar-2013.)
(Revised by Mario Carneiro, 8-Oct-2016.)
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Theorem | r19.23 2468 |
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 2469* |
Theorem 19.23 of [Margaris] p. 90 with
restricted quantifiers.
(Contributed by NM, 31-Aug-1999.)
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Theorem | rexlimi 2470 |
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 2471* |
Inference from Theorem 19.23 of [Margaris] p.
90. (Restricted
quantifier version.) (Contributed by NM, 20-Nov-1994.)
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Theorem | rexlimiva 2472* |
Inference from Theorem 19.23 of [Margaris] p.
90 (restricted quantifier
version). (Contributed by NM, 18-Dec-2006.)
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Theorem | rexlimivw 2473* |
Weaker version of rexlimiv 2471. (Contributed by FL, 19-Sep-2011.)
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Theorem | rexlimd 2474 |
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 2475 |
Version of rexlimd 2474 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 2476* |
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 2477* |
Inference from Theorem 19.23 of [Margaris] p.
90 (restricted quantifier
version). (Contributed by NM, 20-Jan-2007.)
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Theorem | rexlimdvaa 2478* |
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 2479* |
Inference from Theorem 19.23 of [Margaris] p.
90 (restricted quantifier
version). Frequently-used variant of rexlimdv 2476. (Contributed by NM,
7-Jun-2015.)
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Theorem | rexlimdvw 2480* |
Inference from Theorem 19.23 of [Margaris] p.
90 (restricted quantifier
version). (Contributed by NM, 18-Jun-2014.)
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Theorem | rexlimddv 2481* |
Restricted existential elimination rule of natural deduction.
(Contributed by Mario Carneiro, 15-Jun-2016.)
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Theorem | rexlimivv 2482* |
Inference from Theorem 19.23 of [Margaris] p.
90 (restricted quantifier
version). (Contributed by NM, 17-Feb-2004.)
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Theorem | rexlimdvv 2483* |
Inference from Theorem 19.23 of [Margaris] p.
90. (Restricted
quantifier version.) (Contributed by NM, 22-Jul-2004.)
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Theorem | rexlimdvva 2484* |
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 2485 |
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 | r19.26-2 2486 |
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 2487 |
Theorem 19.26 of [Margaris] p. 90 with 3
restricted quantifiers.
(Contributed by FL, 22-Nov-2010.)
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Theorem | r19.26m 2488 |
Theorem 19.26 of [Margaris] p. 90 with mixed
quantifiers. (Contributed by
NM, 22-Feb-2004.)
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Theorem | ralbi 2489 |
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 2490 |
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 2491 |
Split a biconditional and distribute quantifier. (Contributed by NM,
3-Jun-2012.)
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Theorem | r19.27av 2492* |
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 2493* |
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 2494 |
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 2495 |
Variation of Theorem 19.29 of [Margaris] p. 90
with restricted
quantifiers. (Contributed by NM, 31-Aug-1999.)
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Theorem | r19.29af2 2496 |
A commonly used pattern based on r19.29 2494 (Contributed by Thierry
Arnoux, 17-Dec-2017.)
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Theorem | r19.29af 2497* |
A commonly used pattern based on r19.29 2494 (Contributed by Thierry
Arnoux, 29-Nov-2017.)
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Theorem | r19.29a 2498* |
A commonly used pattern based on r19.29 2494 (Contributed by Thierry
Arnoux, 22-Nov-2017.)
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Theorem | r19.29d2r 2499 |
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 2500* |
A commonly used pattern based on r19.29 2494, version with two restricted
quantifiers. (Contributed by Thierry Arnoux, 26-Nov-2017.)
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