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Theorem csbifg 3769
 Description: Distribute proper substitution through the conditional operator. (Contributed by NM, 24-Feb-2013.) (Revised by Mario Carneiro, 14-Nov-2016.)
Assertion
Ref Expression
csbifg

Proof of Theorem csbifg
Dummy variable is distinct from all other variables.
StepHypRef Expression
1 csbeq1 3256 . . 3
2 dfsbcq2 3166 . . . 4
3 csbeq1 3256 . . . 4
4 csbeq1 3256 . . . 4
52, 3, 4ifbieq12d 3763 . . 3
61, 5eqeq12d 2452 . 2
7 vex 2961 . . 3
8 nfs1v 2184 . . . 4
9 nfcsb1v 3285 . . . 4
10 nfcsb1v 3285 . . . 4
118, 9, 10nfif 3765 . . 3
12 sbequ12 1945 . . . 4
13 csbeq1a 3261 . . . 4
14 csbeq1a 3261 . . . 4
1512, 13, 14ifbieq12d 3763 . . 3
167, 11, 15csbief 3294 . 2
176, 16vtoclg 3013 1
 Colors of variables: wff set class Syntax hints:   wi 4   wceq 1653  wsb 1659   wcel 1726  wsbc 3163  csb 3253  cif 3741 This theorem is referenced by:  cdlemk40  31716 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-rab 2716  df-v 2960  df-sbc 3164  df-csb 3254  df-un 3327  df-if 3742
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