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Theorem bnj1000 29314
 Description: Technical lemma for bnj852 29294. 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
bnj1000.1
bnj1000.2
bnj1000.3
bnj1000.15
bnj1000.16
Assertion
Ref Expression
bnj1000
Distinct variable groups:   ,   ,   ,   ,   ,,   ,
Allowed substitution hints:   (,,,,)   (,,,)   (,,,,)   (,,,)   (,,,)   (,,,)   (,,,,)

Proof of Theorem bnj1000
Dummy variables are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 bnj1000.2 . 2
2 df-ral 2712 . . . . 5
32bicomi 195 . . . 4
43sbcbii 3218 . . 3
5 bnj1000.3 . . . . . . 7
6 nfv 1630 . . . . . . . 8
76sbc19.21g 3227 . . . . . . 7
85, 7ax-mp 8 . . . . . 6
9 nfv 1630 . . . . . . . . . 10
109sbc19.21g 3227 . . . . . . . . 9
115, 10ax-mp 8 . . . . . . . 8
12 fveq1 5729 . . . . . . . . . . 11
13 fveq1 5729 . . . . . . . . . . . 12
14 ax-17 1627 . . . . . . . . . . . . 13
15 bnj1000.16 . . . . . . . . . . . . . . . 16
16 nfcv 2574 . . . . . . . . . . . . . . . . 17
17 nfcv 2574 . . . . . . . . . . . . . . . . . . 19
18 bnj1000.15 . . . . . . . . . . . . . . . . . . . 20
19 nfiu1 4123 . . . . . . . . . . . . . . . . . . . 20
2018, 19nfcxfr 2571 . . . . . . . . . . . . . . . . . . 19
2117, 20nfop 4002 . . . . . . . . . . . . . . . . . 18
2221nfsn 3868 . . . . . . . . . . . . . . . . 17
2316, 22nfun 3505 . . . . . . . . . . . . . . . 16
2415, 23nfcxfr 2571 . . . . . . . . . . . . . . 15
25 nfcv 2574 . . . . . . . . . . . . . . 15
2624, 25nffv 5737 . . . . . . . . . . . . . 14
2726nfcrii 2567 . . . . . . . . . . . . 13
2814, 27bnj1316 29194 . . . . . . . . . . . 12
2913, 28syl 16 . . . . . . . . . . 11
3012, 29eqeq12d 2452 . . . . . . . . . 10
31 fveq1 5729 . . . . . . . . . . 11
32 fveq1 5729 . . . . . . . . . . . 12
33 ax-17 1627 . . . . . . . . . . . . 13
3433bnj956 29149 . . . . . . . . . . . 12
3532, 34syl 16 . . . . . . . . . . 11
3631, 35eqeq12d 2452 . . . . . . . . . 10
37 fveq1 5729 . . . . . . . . . . 11
38 fveq1 5729 . . . . . . . . . . . 12
39 ax-17 1627 . . . . . . . . . . . . 13
4039, 27bnj1316 29194 . . . . . . . . . . . 12
4138, 40syl 16 . . . . . . . . . . 11
4237, 41eqeq12d 2452 . . . . . . . . . 10
435, 30, 36, 42bnj610 29117 . . . . . . . . 9
4443imbi2i 305 . . . . . . . 8
4511, 44bitri 242 . . . . . . 7
4645imbi2i 305 . . . . . 6
478, 46bitri 242 . . . . 5
4847albii 1576 . . . 4
49 sbcal 3210 . . . 4
50 df-ral 2712 . . . 4
5148, 49, 503bitr4ri 271 . . 3
52 bnj1000.1 . . . 4
5352sbcbii 3218 . . 3
544, 51, 533bitr4ri 271 . 2
551, 54bitri 242 1
 Colors of variables: wff set class Syntax hints:   wi 4   wb 178  wal 1550   wceq 1653   wcel 1726  wral 2707  cvv 2958  wsbc 3163   cun 3320  csn 3816  cop 3819  ciun 4095   csuc 4585  com 4847  cfv 5456   c-bnj14 29054 This theorem is referenced by:  bnj965  29315 This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1556  ax-5 1567  ax-17 1627  ax-9 1667  ax-8 1688  ax-6 1745  ax-7 1750  ax-11 1762  ax-12 1951  ax-ext 2419 This theorem depends on definitions:  df-bi 179  df-or 361  df-an 362  df-3an 939  df-tru 1329  df-ex 1552  df-nf 1555  df-sb 1660  df-clab 2425  df-cleq 2431  df-clel 2434  df-nfc 2563  df-ral 2712  df-rex 2713  df-rab 2716  df-v 2960  df-sbc 3164  df-dif 3325  df-un 3327  df-in 3329  df-ss 3336  df-nul 3631  df-if 3742  df-sn 3822  df-pr 3823  df-op 3825  df-uni 4018  df-iun 4097  df-br 4215  df-iota 5420  df-fv 5464
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