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Theorem sbcbrg 3813
 Description: Move substitution in and out of a binary relation. (Contributed by NM, 13-Dec-2005.) (Proof shortened by Andrew Salmon, 9-Jul-2011.)
Assertion
Ref Expression
sbcbrg

Proof of Theorem sbcbrg
Dummy variable is distinct from all other variables.
StepHypRef Expression
1 dfsbcq2 2767 . 2
2 csbeq1 2855 . . 3
3 csbeq1 2855 . . 3
4 csbeq1 2855 . . 3
52, 3, 4breq123d 3778 . 2
6 nfcsb1v 2882 . . . 4
7 nfcsb1v 2882 . . . 4
8 nfcsb1v 2882 . . . 4
96, 7, 8nfbr 3808 . . 3
10 csbeq1a 2860 . . . 4
11 csbeq1a 2860 . . . 4
12 csbeq1a 2860 . . . 4
1310, 11, 12breq123d 3778 . . 3
149, 13sbie 1674 . 2
151, 5, 14vtoclbg 2614 1
 Colors of variables: wff set class Syntax hints:   wi 4   wb 98   wceq 1243   wcel 1393  wsb 1645  wsbc 2764  csb 2852   class class class wbr 3764 This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 99  ax-ia2 100  ax-ia3 101  ax-io 630  ax-5 1336  ax-7 1337  ax-gen 1338  ax-ie1 1382  ax-ie2 1383  ax-8 1395  ax-10 1396  ax-11 1397  ax-i12 1398  ax-bndl 1399  ax-4 1400  ax-17 1419  ax-i9 1423  ax-ial 1427  ax-i5r 1428  ax-ext 2022 This theorem depends on definitions:  df-bi 110  df-3an 887  df-tru 1246  df-nf 1350  df-sb 1646  df-clab 2027  df-cleq 2033  df-clel 2036  df-nfc 2167  df-v 2559  df-sbc 2765  df-csb 2853  df-un 2922  df-sn 3381  df-pr 3382  df-op 3384  df-br 3765 This theorem is referenced by:  sbcbr12g  3814  csbcnvg  4519  sbcfung  4925  csbfv12g  5209
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