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Theorem csbifg 3606
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  |-  ( A  e.  V  ->  [_ A  /  x ]_ if (
ph ,  B ,  C )  =  if ( [. A  /  x ]. ph ,  [_ A  /  x ]_ B ,  [_ A  /  x ]_ C ) )

Proof of Theorem csbifg
Dummy variable  y is distinct from all other variables.
StepHypRef Expression
1 csbeq1 3097 . . 3  |-  ( y  =  A  ->  [_ y  /  x ]_ if (
ph ,  B ,  C )  =  [_ A  /  x ]_ if ( ph ,  B ,  C ) )
2 dfsbcq2 3007 . . . 4  |-  ( y  =  A  ->  ( [ y  /  x ] ph  <->  [. A  /  x ]. ph ) )
3 csbeq1 3097 . . . 4  |-  ( y  =  A  ->  [_ y  /  x ]_ B  = 
[_ A  /  x ]_ B )
4 csbeq1 3097 . . . 4  |-  ( y  =  A  ->  [_ y  /  x ]_ C  = 
[_ A  /  x ]_ C )
52, 3, 4ifbieq12d 3600 . . 3  |-  ( y  =  A  ->  if ( [ y  /  x ] ph ,  [_ y  /  x ]_ B ,  [_ y  /  x ]_ C )  =  if ( [. A  /  x ]. ph ,  [_ A  /  x ]_ B ,  [_ A  /  x ]_ C ) )
61, 5eqeq12d 2310 . 2  |-  ( y  =  A  ->  ( [_ y  /  x ]_ if ( ph ,  B ,  C )  =  if ( [ y  /  x ] ph ,  [_ y  /  x ]_ B ,  [_ y  /  x ]_ C )  <->  [_ A  /  x ]_ if ( ph ,  B ,  C )  =  if ( [. A  /  x ]. ph ,  [_ A  /  x ]_ B ,  [_ A  /  x ]_ C ) ) )
7 vex 2804 . . 3  |-  y  e. 
_V
8 nfs1v 2058 . . . 4  |-  F/ x [ y  /  x ] ph
9 nfcsb1v 3126 . . . 4  |-  F/_ x [_ y  /  x ]_ B
10 nfcsb1v 3126 . . . 4  |-  F/_ x [_ y  /  x ]_ C
118, 9, 10nfif 3602 . . 3  |-  F/_ x if ( [ y  /  x ] ph ,  [_ y  /  x ]_ B ,  [_ y  /  x ]_ C )
12 sbequ12 1872 . . . 4  |-  ( x  =  y  ->  ( ph 
<->  [ y  /  x ] ph ) )
13 csbeq1a 3102 . . . 4  |-  ( x  =  y  ->  B  =  [_ y  /  x ]_ B )
14 csbeq1a 3102 . . . 4  |-  ( x  =  y  ->  C  =  [_ y  /  x ]_ C )
1512, 13, 14ifbieq12d 3600 . . 3  |-  ( x  =  y  ->  if ( ph ,  B ,  C )  =  if ( [ y  /  x ] ph ,  [_ y  /  x ]_ B ,  [_ y  /  x ]_ C ) )
167, 11, 15csbief 3135 . 2  |-  [_ y  /  x ]_ if (
ph ,  B ,  C )  =  if ( [ y  /  x ] ph ,  [_ y  /  x ]_ B ,  [_ y  /  x ]_ C )
176, 16vtoclg 2856 1  |-  ( A  e.  V  ->  [_ A  /  x ]_ if (
ph ,  B ,  C )  =  if ( [. A  /  x ]. ph ,  [_ A  /  x ]_ B ,  [_ A  /  x ]_ C ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    = wceq 1632   [wsb 1638    e. wcel 1696   [.wsbc 3004   [_csb 3094   ifcif 3578
This theorem is referenced by:  cdlemk40  31728
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1536  ax-5 1547  ax-17 1606  ax-9 1644  ax-8 1661  ax-6 1715  ax-7 1720  ax-11 1727  ax-12 1878  ax-ext 2277
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3an 936  df-tru 1310  df-ex 1532  df-nf 1535  df-sb 1639  df-clab 2283  df-cleq 2289  df-clel 2292  df-nfc 2421  df-rab 2565  df-v 2803  df-sbc 3005  df-csb 3095  df-un 3170  df-if 3579
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