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Theorem csbifgOLD 3992
Description: Distribute proper substitution through the conditional operator. (Contributed by NM, 24-Feb-2013.) (Revised by Mario Carneiro, 14-Nov-2016.) Obsolete as of 19-Aug-2018. Use csbif 3991 instead. (New usage is discouraged.) (Proof modification is discouraged.)
Assertion
Ref Expression
csbifgOLD

Proof of Theorem csbifgOLD
Dummy variable is distinct from all other variables.
StepHypRef Expression
1 csbeq1 3437 . . 3
2 dfsbcq2 3330 . . . 4
3 csbeq1 3437 . . . 4
4 csbeq1 3437 . . . 4
52, 3, 4ifbieq12d 3968 . . 3
61, 5eqeq12d 2479 . 2
7 vex 3112 . . 3
8 nfs1v 2181 . . . 4
9 nfcsb1v 3450 . . . 4
10 nfcsb1v 3450 . . . 4
118, 9, 10nfif 3970 . . 3
12 sbequ12 1992 . . . 4
13 csbeq1a 3443 . . . 4
14 csbeq1a 3443 . . . 4
1512, 13, 14ifbieq12d 3968 . . 3
167, 11, 15csbief 3459 . 2
176, 16vtoclg 3167 1
Colors of variables: wff setvar class
Syntax hints:  ->wi 4  =wceq 1395  [wsb 1739  e.wcel 1818  [.wsbc 3327  [_csb 3434  ifcif 3941
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-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-un 3480  df-if 3942
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