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Theorem fvif 5540
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 5525 . 2  |-  ( if ( ph ,  A ,  B )  =  A  ->  ( F `  if ( ph ,  A ,  B ) )  =  ( F `  A
) )
2 fveq2 5525 . 2  |-  ( if ( ph ,  A ,  B )  =  B  ->  ( F `  if ( ph ,  A ,  B ) )  =  ( F `  B
) )
31, 2ifsb 3574 1  |-  ( F `
 if ( ph ,  A ,  B ) )  =  if (
ph ,  ( F `
 A ) ,  ( F `  B
) )
Colors of variables: wff set class
Syntax hints:    = wceq 1623   ifcif 3565   ` cfv 5255
This theorem is referenced by:  ccatco  11490  sumeq2ii  12166  ruclem1  12509  xpslem  13475  copco  18516  pcopt  18520  pcopt2  18521  limccnp  19241  prmorcht  20416  pclogsum  20454  fvifOLD  26376
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1533  ax-5 1544  ax-17 1603  ax-9 1635  ax-8 1643  ax-6 1703  ax-7 1708  ax-11 1715  ax-12 1866  ax-ext 2264
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3an 936  df-tru 1310  df-ex 1529  df-nf 1532  df-sb 1630  df-clab 2270  df-cleq 2276  df-clel 2279  df-nfc 2408  df-rex 2549  df-rab 2552  df-v 2790  df-dif 3155  df-un 3157  df-in 3159  df-ss 3166  df-nul 3456  df-if 3566  df-sn 3646  df-pr 3647  df-op 3649  df-uni 3828  df-br 4024  df-iota 5219  df-fv 5263
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