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Theorem bnj1312 33549
 Description: Technical lemma for bnj60 33553. This lemma may no longer be used or have become an indirect lemma of the theorem in question (i.e. a lemma of a lemma... of the theorem). (Contributed by Jonathan Ben-Naim, 3-Jun-2011.) (New usage is discouraged.)
Hypotheses
Ref Expression
bnj1312.1
bnj1312.2
bnj1312.3
bnj1312.4
bnj1312.5
bnj1312.6
bnj1312.7
bnj1312.8
bnj1312.9
bnj1312.10
bnj1312.11
bnj1312.12
bnj1312.13
bnj1312.14
Assertion
Ref Expression
bnj1312
Distinct variable groups:   ,,,,,   ,   ,   ,   ,,,,   ,,,,,   ,   ,,,,,   ,   ,   ,   ,
Allowed substitution hints:   (,,,)   (,,,)   (,,,)   (,,,)   (,,,)   (,,,)   (,,,,)   (,,,)   ()   (,,,,)   (,,,,)   (,,,)   (,,,,)   (,,,,)

Proof of Theorem bnj1312
Dummy variable is distinct from all other variables.
StepHypRef Expression
1 bnj1312.5 . . 3
2 bnj1312.6 . . . 4
32simplbi 460 . . . . . . 7
41bnj21 33206 . . . . . . . 8
54a1i 11 . . . . . . 7
62simprbi 464 . . . . . . 7
71bnj1230 33296 . . . . . . . 8
87bnj1228 33502 . . . . . . 7
93, 5, 6, 8syl3anc 1228 . . . . . 6
10 bnj1312.7 . . . . . 6
11 nfv 1683 . . . . . . . . 9
127nfcii 2619 . . . . . . . . . 10
13 nfcv 2629 . . . . . . . . . 10
1412, 13nfne 2798 . . . . . . . . 9
1511, 14nfan 1875 . . . . . . . 8
162, 15nfxfr 1625 . . . . . . 7
1716nfri 1822 . . . . . 6
189, 10, 17bnj1521 33344 . . . . 5
1910simp2bi 1012 . . . . 5
201bnj1538 33348 . . . . . 6
21 bnj1312.1 . . . . . . . . 9
22 bnj1312.2 . . . . . . . . 9
23 bnj1312.3 . . . . . . . . 9
24 bnj1312.4 . . . . . . . . 9
25 bnj1312.8 . . . . . . . . 9
26 bnj1312.9 . . . . . . . . 9
27 bnj1312.10 . . . . . . . . 9
28 bnj1312.11 . . . . . . . . 9
29 bnj1312.12 . . . . . . . . 9
3021, 22, 23, 24, 1, 2, 10, 25, 26, 27, 28, 29bnj1489 33547 . . . . . . . 8
31 bnj1312.13 . . . . . . . . . . 11
32 bnj1312.14 . . . . . . . . . . 11
3310, 3bnj835 33252 . . . . . . . . . . . . . 14
3421, 22, 23, 24, 1, 2, 10, 25, 26, 27bnj1384 33523 . . . . . . . . . . . . . 14
3533, 34syl 16 . . . . . . . . . . . . 13
3621, 22, 23, 24, 1, 2, 10, 25, 26, 27bnj1415 33529 . . . . . . . . . . . . 13
3735, 36bnj1422 33331 . . . . . . . . . . . 12
3821, 22, 23, 24, 1, 2, 10, 25, 26, 27, 28, 29, 36bnj1416 33530 . . . . . . . . . . . . . 14
3921, 22, 23, 24, 1, 2, 10, 25, 26, 27, 28, 29, 35, 38, 36bnj1421 33533 . . . . . . . . . . . . 13
4039, 38bnj1422 33331 . . . . . . . . . . . 12
4121, 22, 23, 24, 1, 2, 10, 25, 26, 27, 28, 29, 31, 32, 37, 40bnj1423 33542 . . . . . . . . . . 11
4232fneq2i 5682 . . . . . . . . . . . 12
4340, 42sylibr 212 . . . . . . . . . . 11
4421, 22, 23, 24, 1, 2, 10, 25, 26, 27, 28, 29, 31, 32bnj1452 33543 . . . . . . . . . . 11
4521, 22, 23, 24, 1, 2, 10, 25, 26, 27, 28, 29, 31, 32, 30, 41, 43, 44bnj1463 33546 . . . . . . . . . 10
4645, 38jca 532 . . . . . . . . 9
4721, 22, 23, 24, 1, 2, 10, 25, 26, 27, 28, 29, 46bnj1491 33548 . . . . . . . 8
4830, 47mpdan 668 . . . . . . 7
4948, 24bnj1198 33289 . . . . . 6
5020, 49nsyl3 119 . . . . 5
5118, 19, 50bnj1304 33313 . . . 4
522, 51bnj1541 33349 . . 3
531, 52bnj1476 33340 . 2
5424exbii 1644 . . . 4
55 df-rex 2823 . . . 4
5654, 55bitr4i 252 . . 3
5756ralbii 2898 . 2
5853, 57sylib 196 1
 Colors of variables: wff setvar class Syntax hints:   wn 3   wi 4   wb 184   wa 369   w3a 973   wceq 1379  wex 1596   wcel 1767  cab 2452   wne 2662  wral 2817  wrex 2818  crab 2821  cvv 3118  wsbc 3336   cun 3479   wss 3481  c0 3790  csn 4033  cop 4039  cuni 4251   class class class wbr 4453   cdm 5005   cres 5007   wfun 5588   wfn 5589  cfv 5594   c-bnj14 33176   w-bnj15 33180   c-bnj18 33182 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1601  ax-4 1612  ax-5 1680  ax-6 1719  ax-7 1739  ax-8 1769  ax-9 1771  ax-10 1786  ax-11 1791  ax-12 1803  ax-13 1968  ax-ext 2445  ax-rep 4564  ax-sep 4574  ax-nul 4582  ax-pow 4631  ax-pr 4692  ax-un 6587  ax-reg 8030  ax-inf2 8070 This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 974  df-3an 975  df-tru 1382  df-fal 1385  df-ex 1597  df-nf 1600  df-sb 1712  df-eu 2279  df-mo 2280  df-clab 2453  df-cleq 2459  df-clel 2462  df-nfc 2617  df-ne 2664  df-ral 2822  df-rex 2823  df-reu 2824  df-rab 2826  df-v 3120  df-sbc 3337  df-csb 3441  df-dif 3484  df-un 3486  df-in 3488  df-ss 3495  df-pss 3497  df-nul 3791  df-if 3946  df-pw 4018  df-sn 4034  df-pr 4036  df-tp 4038  df-op 4040  df-uni 4252  df-iun 4333  df-br 4454  df-opab 4512  df-mpt 4513  df-tr 4547  df-eprel 4797  df-id 4801  df-po 4806  df-so 4807  df-fr 4844  df-we 4846  df-ord 4887  df-on 4888  df-lim 4889  df-suc 4890  df-xp 5011  df-rel 5012  df-cnv 5013  df-co 5014  df-dm 5015  df-rn 5016  df-res 5017  df-ima 5018  df-iota 5557  df-fun 5596  df-fn 5597  df-f 5598  df-f1 5599  df-fo 5600  df-f1o 5601  df-fv 5602  df-om 6696  df-1o 7142  df-bnj17 33175  df-bnj14 33177  df-bnj13 33179  df-bnj15 33181  df-bnj18 33183  df-bnj19 33185 This theorem is referenced by:  bnj1493  33550
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