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Theorem bnj1523 29880
 Description: Technical lemma for bnj1522 29881. 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
bnj1523.1
bnj1523.2
bnj1523.3
bnj1523.4
bnj1523.5
bnj1523.6
bnj1523.7
bnj1523.8
bnj1523.9
Assertion
Ref Expression
bnj1523
Distinct variable groups:   ,,,   ,,,   ,   ,,   ,,   ,,,   ,   ,,,   ,,,   ,,   ,   ,
Allowed substitution hints:   (,,,,)   (,,,,)   (,,,)   (,,,,)   (,,,)   (,,,,)   (,,)   (,,)   ()   (,)   (,,,)

Proof of Theorem bnj1523
Dummy variables are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 bnj1523.5 . 2
2 bnj1523.6 . . 3
3 bnj1523.9 . . . . . . . . . . . . 13
4 bnj1523.7 . . . . . . . . . . . . . 14
5 bnj1523.1 . . . . . . . . . . . . . . . . 17
6 bnj1523.2 . . . . . . . . . . . . . . . . 17
7 bnj1523.3 . . . . . . . . . . . . . . . . 17
8 bnj1523.4 . . . . . . . . . . . . . . . . 17
95, 6, 7, 8bnj60 29871 . . . . . . . . . . . . . . . 16
101, 9bnj835 29570 . . . . . . . . . . . . . . 15
112, 10bnj832 29568 . . . . . . . . . . . . . 14
124, 11bnj835 29570 . . . . . . . . . . . . 13
133, 12bnj835 29570 . . . . . . . . . . . 12
141simp2bi 1024 . . . . . . . . . . . . . . 15
152, 14bnj832 29568 . . . . . . . . . . . . . 14
164, 15bnj835 29570 . . . . . . . . . . . . 13
173, 16bnj835 29570 . . . . . . . . . . . 12
18 bnj213 29693 . . . . . . . . . . . . 13
1918a1i 11 . . . . . . . . . . . 12
203simp3bi 1025 . . . . . . . . . . . . . . . . 17
2120bnj1211 29609 . . . . . . . . . . . . . . . 16
22 con2b 336 . . . . . . . . . . . . . . . . 17
2322albii 1691 . . . . . . . . . . . . . . . 16
2421, 23sylib 200 . . . . . . . . . . . . . . 15
25 bnj1418 29849 . . . . . . . . . . . . . . . . 17
2625imim1i 60 . . . . . . . . . . . . . . . 16
2726alimi 1684 . . . . . . . . . . . . . . 15
2824, 27syl 17 . . . . . . . . . . . . . 14
2928bnj1142 29601 . . . . . . . . . . . . 13
30 bnj1523.8 . . . . . . . . . . . . . 14
315bnj1309 29831 . . . . . . . . . . . . . . . . . . 19
327, 31bnj1307 29832 . . . . . . . . . . . . . . . . . 18
3332nfcii 2583 . . . . . . . . . . . . . . . . 17
3433nfuni 4204 . . . . . . . . . . . . . . . 16
358, 34nfcxfr 2590 . . . . . . . . . . . . . . 15
3635nfcrii 2585 . . . . . . . . . . . . . 14
3730, 36bnj1534 29664 . . . . . . . . . . . . 13
3829, 18, 37bnj1533 29663 . . . . . . . . . . . 12
3913, 17, 19, 38bnj1536 29665 . . . . . . . . . . 11
4039opeq2d 4173 . . . . . . . . . 10
4140fveq2d 5869 . . . . . . . . 9
425, 6, 7, 8bnj1500 29877 . . . . . . . . . . . . . . 15
431, 42bnj835 29570 . . . . . . . . . . . . . 14
442, 43bnj832 29568 . . . . . . . . . . . . 13
454, 44bnj835 29570 . . . . . . . . . . . 12
4645, 36bnj1529 29879 . . . . . . . . . . 11
473, 46bnj835 29570 . . . . . . . . . 10
4830bnj21 29523 . . . . . . . . . . 11
493simp2bi 1024 . . . . . . . . . . 11
5048, 49bnj1213 29610 . . . . . . . . . 10
5147, 50bnj1294 29629 . . . . . . . . 9
521simp3bi 1025 . . . . . . . . . . . . . 14
532, 52bnj832 29568 . . . . . . . . . . . . 13
544, 53bnj835 29570 . . . . . . . . . . . 12
55 ax-5 1758 . . . . . . . . . . . 12
5654, 55bnj1529 29879 . . . . . . . . . . 11
573, 56bnj835 29570 . . . . . . . . . 10
5857, 50bnj1294 29629 . . . . . . . . 9
5941, 51, 583eqtr4d 2495 . . . . . . . 8
6030, 36bnj1534 29664 . . . . . . . . . . 11
6160bnj1538 29666 . . . . . . . . . 10
623, 61bnj836 29571 . . . . . . . . 9
6362neneqd 2629 . . . . . . . 8
6459, 63pm2.65i 177 . . . . . . 7
6564nex 1678 . . . . . 6
661simp1bi 1023 . . . . . . . . . 10
672, 66bnj832 29568 . . . . . . . . 9
684, 67bnj835 29570 . . . . . . . 8
6948a1i 11 . . . . . . . 8
704simp2bi 1024 . . . . . . . . . 10
714simp3bi 1025 . . . . . . . . . 10
7230rabeq2i 3042 . . . . . . . . . 10
7370, 71, 72sylanbrc 670 . . . . . . . . 9
74 ne0i 3737 . . . . . . . . 9
7573, 74syl 17 . . . . . . . 8
76 bnj69 29819 . . . . . . . 8
7768, 69, 75, 76syl3anc 1268 . . . . . . 7
7877, 3bnj1209 29608 . . . . . 6
7965, 78mto 180 . . . . 5
8079nex 1678 . . . 4
812simprbi 466 . . . . . 6
8211, 15, 81, 36bnj1542 29668 . . . . 5
835, 6, 7, 8, 1, 2bnj1525 29878 . . . . 5
8482, 4, 83bnj1521 29662 . . . 4
8580, 84mto 180 . . 3
862, 85bnj1541 29667 . 2
871, 86sylbir 217 1
 Colors of variables: wff setvar class Syntax hints:   wn 3   wi 4   wb 188   wa 371   w3a 985  wal 1442   wceq 1444  wex 1663   wcel 1887  cab 2437   wne 2622  wral 2737  wrex 2738  crab 2741   wss 3404  c0 3731  cop 3974  cuni 4198   class class class wbr 4402   cres 4836   wfn 5577  cfv 5582   c-bnj14 29493   w-bnj15 29497 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1669  ax-4 1682  ax-5 1758  ax-6 1805  ax-7 1851  ax-8 1889  ax-9 1896  ax-10 1915  ax-11 1920  ax-12 1933  ax-13 2091  ax-ext 2431  ax-rep 4515  ax-sep 4525  ax-nul 4534  ax-pow 4581  ax-pr 4639  ax-un 6583  ax-reg 8107  ax-inf2 8146 This theorem depends on definitions:  df-bi 189  df-or 372  df-an 373  df-3or 986  df-3an 987  df-tru 1447  df-fal 1450  df-ex 1664  df-nf 1668  df-sb 1798  df-eu 2303  df-mo 2304  df-clab 2438  df-cleq 2444  df-clel 2447  df-nfc 2581  df-ne 2624  df-ral 2742  df-rex 2743  df-reu 2744  df-rab 2746  df-v 3047  df-sbc 3268  df-csb 3364  df-dif 3407  df-un 3409  df-in 3411  df-ss 3418  df-pss 3420  df-nul 3732  df-if 3882  df-pw 3953  df-sn 3969  df-pr 3971  df-tp 3973  df-op 3975  df-uni 4199  df-iun 4280  df-br 4403  df-opab 4462  df-mpt 4463  df-tr 4498  df-eprel 4745  df-id 4749  df-po 4755  df-so 4756  df-fr 4793  df-we 4795  df-xp 4840  df-rel 4841  df-cnv 4842  df-co 4843  df-dm 4844  df-rn 4845  df-res 4846  df-ima 4847  df-ord 5426  df-on 5427  df-lim 5428  df-suc 5429  df-iota 5546  df-fun 5584  df-fn 5585  df-f 5586  df-f1 5587  df-fo 5588  df-f1o 5589  df-fv 5590  df-om 6693  df-1o 7182  df-bnj17 29492  df-bnj14 29494  df-bnj13 29496  df-bnj15 29498  df-bnj18 29500  df-bnj19 29502 This theorem is referenced by:  bnj1522  29881
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