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Theorem sbcbr123 4503
Description: Move substitution in and out of a binary relation. (Contributed by NM, 13-Dec-2005.) (Modified by NM, 22-Aug-2018.)
Assertion
Ref Expression
sbcbr123

Proof of Theorem sbcbr123
Dummy variable is distinct from all other variables.
StepHypRef Expression
1 dfsbcq2 3330 . . 3
2 csbeq1 3437 . . . 4
3 csbeq1 3437 . . . 4
4 csbeq1 3437 . . . 4
52, 3, 4breq123d 4466 . . 3
6 nfcsb1v 3450 . . . . 5
7 nfcsb1v 3450 . . . . 5
8 nfcsb1v 3450 . . . . 5
96, 7, 8nfbr 4496 . . . 4
10 csbeq1a 3443 . . . . 5
11 csbeq1a 3443 . . . . 5
12 csbeq1a 3443 . . . . 5
1310, 11, 12breq123d 4466 . . . 4
149, 13sbie 2149 . . 3
151, 5, 14vtoclbg 3168 . 2
16 sbcex 3337 . . . 4
1716con3i 135 . . 3
18 br0 4498 . . . 4
19 csbprc 3821 . . . . 5
2019breqd 4463 . . . 4
2118, 20mtbiri 303 . . 3
2217, 212falsed 351 . 2
2315, 22pm2.61i 164 1
Colors of variables: wff setvar class
Syntax hints:  -.wn 3  <->wb 184  =wceq 1395  [wsb 1739  e.wcel 1818   cvv 3109  [.wsbc 3327  [_csb 3434   c0 3784   class class class wbr 4452
This theorem is referenced by:  sbcbr  4505  sbcbr12g  4506  csbcnvgALT  5192  sbcfung  5616  csbfv12  5906
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1618  ax-4 1631  ax-5 1704  ax-6 1747  ax-7 1790  ax-10 1837  ax-11 1842  ax-12 1854  ax-13 1999  ax-ext 2435
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 975  df-tru 1398  df-fal 1401  df-ex 1613  df-nf 1617  df-sb 1740  df-clab 2443  df-cleq 2449  df-clel 2452  df-nfc 2607  df-rab 2816  df-v 3111  df-sbc 3328  df-csb 3435  df-dif 3478  df-un 3480  df-in 3482  df-ss 3489  df-nul 3785  df-if 3942  df-sn 4030  df-pr 4032  df-op 4036  df-br 4453
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