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Theorem csbif 3991
Description: Distribute proper substitution through the conditional operator. (Contributed by NM, 24-Feb-2013.) (Revised by NM, 19-Aug-2018.)
Assertion
Ref Expression
csbif

Proof of Theorem csbif
Dummy variable is distinct from all other variables.
StepHypRef Expression
1 csbeq1 3437 . . . 4
2 dfsbcq2 3330 . . . . 5
3 csbeq1 3437 . . . . 5
4 csbeq1 3437 . . . . 5
52, 3, 4ifbieq12d 3968 . . . 4
61, 5eqeq12d 2479 . . 3
7 vex 3112 . . . 4
8 nfs1v 2181 . . . . 5
9 nfcsb1v 3450 . . . . 5
10 nfcsb1v 3450 . . . . 5
118, 9, 10nfif 3970 . . . 4
12 sbequ12 1992 . . . . 5
13 csbeq1a 3443 . . . . 5
14 csbeq1a 3443 . . . . 5
1512, 13, 14ifbieq12d 3968 . . . 4
167, 11, 15csbief 3459 . . 3
176, 16vtoclg 3167 . 2
18 csbprc 3821 . . 3
19 csbprc 3821 . . . . 5
20 csbprc 3821 . . . . 5
2119, 20ifeq12d 3961 . . . 4
22 ifid 3978 . . . 4
2321, 22syl6req 2515 . . 3
2418, 23eqtrd 2498 . 2
2517, 24pm2.61i 164 1
Colors of variables: wff setvar class
Syntax hints:  -.wn 3  =wceq 1395  [wsb 1739  e.wcel 1818   cvv 3109  [.wsbc 3327  [_csb 3434   c0 3784  ifcif 3941
This theorem is referenced by:  fvmptnn04if  19350  cdlemk40  36643
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
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