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Theorem bnj1523 29417
 Description: Technical lemma for bnj1522 29418. 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 . . . . . . . . . . 11
4 bnj1523.7 . . . . . . . . . . . . 13
5 bnj1523.1 . . . . . . . . . . . . . . . 16
6 bnj1523.2 . . . . . . . . . . . . . . . 16
7 bnj1523.3 . . . . . . . . . . . . . . . 16
8 bnj1523.4 . . . . . . . . . . . . . . . 16
95, 6, 7, 8bnj1500 29414 . . . . . . . . . . . . . . 15
101, 9bnj835 29105 . . . . . . . . . . . . . 14
112, 10bnj832 29103 . . . . . . . . . . . . 13
124, 11bnj835 29105 . . . . . . . . . . . 12
135bnj1309 29368 . . . . . . . . . . . . . . . . 17
147, 13bnj1307 29369 . . . . . . . . . . . . . . . 16
1514nfcii 2423 . . . . . . . . . . . . . . 15
1615nfuni 3849 . . . . . . . . . . . . . 14
178, 16nfcxfr 2429 . . . . . . . . . . . . 13
1817nfcrii 2425 . . . . . . . . . . . 12
1912, 18bnj1529 29416 . . . . . . . . . . 11
203, 19bnj835 29105 . . . . . . . . . 10
21 bnj1523.8 . . . . . . . . . . . 12
2221bnj21 29059 . . . . . . . . . . 11
233simp2bi 971 . . . . . . . . . . 11
2422, 23bnj1213 29147 . . . . . . . . . 10
2520, 24bnj1294 29166 . . . . . . . . 9
261simp3bi 972 . . . . . . . . . . . . . . 15
272, 26bnj832 29103 . . . . . . . . . . . . . 14
284, 27bnj835 29105 . . . . . . . . . . . . 13
29 ax-17 1606 . . . . . . . . . . . . 13
3028, 29bnj1529 29416 . . . . . . . . . . . 12
313, 30bnj835 29105 . . . . . . . . . . 11
3231, 24bnj1294 29166 . . . . . . . . . 10
335, 6, 7, 8bnj60 29408 . . . . . . . . . . . . . . . . 17
341, 33bnj835 29105 . . . . . . . . . . . . . . . 16
352, 34bnj832 29103 . . . . . . . . . . . . . . 15
364, 35bnj835 29105 . . . . . . . . . . . . . 14
373, 36bnj835 29105 . . . . . . . . . . . . 13
381simp2bi 971 . . . . . . . . . . . . . . . 16
392, 38bnj832 29103 . . . . . . . . . . . . . . 15
404, 39bnj835 29105 . . . . . . . . . . . . . 14
413, 40bnj835 29105 . . . . . . . . . . . . 13
42 bnj213 29230 . . . . . . . . . . . . . 14
4342a1i 10 . . . . . . . . . . . . 13
443simp3bi 972 . . . . . . . . . . . . . . . . . 18
4544bnj1211 29146 . . . . . . . . . . . . . . . . 17
46 con2b 324 . . . . . . . . . . . . . . . . . 18
4746albii 1556 . . . . . . . . . . . . . . . . 17
4845, 47sylib 188 . . . . . . . . . . . . . . . 16
49 bnj1418 29386 . . . . . . . . . . . . . . . . . 18
5049imim1i 54 . . . . . . . . . . . . . . . . 17
5150alimi 1549 . . . . . . . . . . . . . . . 16
5248, 51syl 15 . . . . . . . . . . . . . . 15
5352bnj1142 29137 . . . . . . . . . . . . . 14
5421, 18bnj1534 29201 . . . . . . . . . . . . . 14
5553, 42, 54bnj1533 29200 . . . . . . . . . . . . 13
5637, 41, 43, 55bnj1536 29202 . . . . . . . . . . . 12
5756opeq2d 3819 . . . . . . . . . . 11
5857fveq2d 5545 . . . . . . . . . 10
5932, 58eqtr4d 2331 . . . . . . . . 9
6025, 59eqtr4d 2331 . . . . . . . 8
6121, 18bnj1534 29201 . . . . . . . . . . 11
6261bnj1538 29203 . . . . . . . . . 10
633, 62bnj836 29106 . . . . . . . . 9
6463neneqd 2475 . . . . . . . 8
6560, 64pm2.65i 165 . . . . . . 7
6665nex 1545 . . . . . 6
671simp1bi 970 . . . . . . . . . 10
682, 67bnj832 29103 . . . . . . . . 9
694, 68bnj835 29105 . . . . . . . 8
7022a1i 10 . . . . . . . 8
714simp2bi 971 . . . . . . . . . 10
724simp3bi 972 . . . . . . . . . 10
7321rabeq2i 2798 . . . . . . . . . 10
7471, 72, 73sylanbrc 645 . . . . . . . . 9
75 ne0i 3474 . . . . . . . . 9
7674, 75syl 15 . . . . . . . 8
77 bnj69 29356 . . . . . . . 8
7869, 70, 76, 77syl3anc 1182 . . . . . . 7
7978, 3bnj1209 29145 . . . . . 6
8066, 79mto 167 . . . . 5
8180nex 1545 . . . 4
822simprbi 450 . . . . . 6
8335, 39, 82, 18bnj1542 29205 . . . . 5
845, 6, 7, 8, 1, 2bnj1525 29415 . . . . 5
8583, 4, 84bnj1521 29199 . . . 4
8681, 85mto 167 . . 3
872, 86bnj1541 29204 . 2
881, 87sylbir 204 1
 Colors of variables: wff set class Syntax hints:   wn 3   wi 4   wb 176   wa 358   w3a 934  wal 1530  wex 1531   wceq 1632   wcel 1696  cab 2282   wne 2459  wral 2556  wrex 2557  crab 2560   wss 3165  c0 3468  cop 3656  cuni 3843   class class class wbr 4039   cres 4707   wfn 5266  cfv 5271   c-bnj14 29029   w-bnj15 29033 This theorem is referenced by:  bnj1522  29418 This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1536  ax-5 1547  ax-17 1606  ax-9 1644  ax-8 1661  ax-13 1698  ax-14 1700  ax-6 1715  ax-7 1720  ax-11 1727  ax-12 1878  ax-ext 2277  ax-rep 4147  ax-sep 4157  ax-nul 4165  ax-pow 4204  ax-pr 4230  ax-un 4528  ax-reg 7322  ax-inf2 7358 This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3or 935  df-3an 936  df-tru 1310  df-ex 1532  df-nf 1535  df-sb 1639  df-eu 2160  df-mo 2161  df-clab 2283  df-cleq 2289  df-clel 2292  df-nfc 2421  df-ne 2461  df-ral 2561  df-rex 2562  df-reu 2563  df-rab 2565  df-v 2803  df-sbc 3005  df-csb 3095  df-dif 3168  df-un 3170  df-in 3172  df-ss 3179  df-pss 3181  df-nul 3469  df-if 3579  df-pw 3640  df-sn 3659  df-pr 3660  df-tp 3661  df-op 3662  df-uni 3844  df-iun 3923  df-br 4040  df-opab 4094  df-mpt 4095  df-tr 4130  df-eprel 4321  df-id 4325  df-po 4330  df-so 4331  df-fr 4368  df-we 4370  df-ord 4411  df-on 4412  df-lim 4413  df-suc 4414  df-om 4673  df-xp 4711  df-rel 4712  df-cnv 4713  df-co 4714  df-dm 4715  df-rn 4716  df-res 4717  df-ima 4718  df-iota 5235  df-fun 5273  df-fn 5274  df-f 5275  df-f1 5276  df-fo 5277  df-f1o 5278  df-fv 5279  df-1o 6495  df-bnj17 29028  df-bnj14 29030  df-bnj13 29032  df-bnj15 29034  df-bnj18 29036  df-bnj19 29038
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