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Theorem fvif 5744
Description: Move a conditional outside of a function. (Contributed by Jeff Madsen, 2-Sep-2009.)
Assertion
Ref Expression
fvif  |-  ( F `
 if ( ph ,  A ,  B ) )  =  if (
ph ,  ( F `
 A ) ,  ( F `  B
) )

Proof of Theorem fvif
StepHypRef Expression
1 fveq2 5729 . 2  |-  ( if ( ph ,  A ,  B )  =  A  ->  ( F `  if ( ph ,  A ,  B ) )  =  ( F `  A
) )
2 fveq2 5729 . 2  |-  ( if ( ph ,  A ,  B )  =  B  ->  ( F `  if ( ph ,  A ,  B ) )  =  ( F `  B
) )
31, 2ifsb 3749 1  |-  ( F `
 if ( ph ,  A ,  B ) )  =  if (
ph ,  ( F `
 A ) ,  ( F `  B
) )
Colors of variables: wff set class
Syntax hints:    = wceq 1653   ifcif 3740   ` cfv 5455
This theorem is referenced by:  ccatco  11805  sumeq2ii  12488  ruclem1  12831  xpslem  13799  copco  19044  pcopt  19048  pcopt2  19049  limccnp  19779  prmorcht  20962  pclogsum  21000  prodeq2ii  25240  mblfinlem2  26245  ftc1anclem8  26288  ftc1anc  26289  fvifeq  28077
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 2418
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 2424  df-cleq 2430  df-clel 2433  df-nfc 2562  df-rex 2712  df-rab 2715  df-v 2959  df-dif 3324  df-un 3326  df-in 3328  df-ss 3335  df-nul 3630  df-if 3741  df-sn 3821  df-pr 3822  df-op 3824  df-uni 4017  df-br 4214  df-iota 5419  df-fv 5463
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