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Theorem csbifg 3690
 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 3139 . . 3
2 dfsbcq2 3049 . . . 4
3 csbeq1 3139 . . . 4
4 csbeq1 3139 . . . 4
52, 3, 4ifbieq12d 3684 . . 3
61, 5eqeq12d 2367 . 2
7 vex 2862 . . 3
8 nfs1v 2106 . . . 4
9 nfcsb1v 3168 . . . 4
10 nfcsb1v 3168 . . . 4
118, 9, 10nfif 3686 . . 3
12 sbequ12 1919 . . . 4
13 csbeq1a 3144 . . . 4
14 csbeq1a 3144 . . . 4
1512, 13, 14ifbieq12d 3684 . . 3
167, 11, 15csbief 3177 . 2
176, 16vtoclg 2914 1
 Colors of variables: wff setvar class Syntax hints:   wi 4   wceq 1642  wsb 1648   wcel 1710  wsbc 3046  csb 3136  cif 3662 This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1546  ax-5 1557  ax-17 1616  ax-9 1654  ax-8 1675  ax-6 1729  ax-7 1734  ax-11 1746  ax-12 1925  ax-ext 2334 This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3an 936  df-nan 1288  df-tru 1319  df-ex 1542  df-nf 1545  df-sb 1649  df-clab 2340  df-cleq 2346  df-clel 2349  df-nfc 2478  df-rab 2623  df-v 2861  df-sbc 3047  df-csb 3137  df-nin 3211  df-compl 3212  df-un 3214  df-if 3663 This theorem is referenced by: (None)
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