Metamath Proof Explorer < Previous   Next > Nearby theorems Mirrors  >  Home  >  MPE Home  >  Th. List  >  sbcbrg Structured version   Unicode version

Theorem sbcbrg 4262
 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 3165 . 2
2 csbeq1 3255 . . 3
3 csbeq1 3255 . . 3
4 csbeq1 3255 . . 3
52, 3, 4breq123d 4227 . 2
6 nfcsb1v 3284 . . . 4
7 nfcsb1v 3284 . . . 4
8 nfcsb1v 3284 . . . 4
96, 7, 8nfbr 4257 . . 3
10 csbeq1a 3260 . . . 4
11 csbeq1a 3260 . . . 4
12 csbeq1a 3260 . . . 4
1310, 11, 12breq123d 4227 . . 3
149, 13sbie 2150 . 2
151, 5, 14vtoclbg 3013 1
 Colors of variables: wff set class Syntax hints:   wi 4   wb 178   wceq 1653  wsb 1659   wcel 1726  wsbc 3162  csb 3252   class class class wbr 4213 This theorem is referenced by:  sbcbr12g  4263  csbfv12g  5739  csbcnvg  24038  sbcfun  27964 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 2418 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 2424  df-cleq 2430  df-clel 2433  df-nfc 2562  df-rab 2715  df-v 2959  df-sbc 3163  df-csb 3253  df-dif 3324  df-un 3326  df-in 3328  df-ss 3335  df-nul 3630  df-if 3741  df-sn 3821  df-pr 3822  df-op 3824  df-br 4214
 Copyright terms: Public domain W3C validator