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Theorem ifsb 3800
Description: Distribute a function over an if-clause. (Contributed by Mario Carneiro, 14-Aug-2013.)
Hypotheses
Ref Expression
ifsb.1  |-  ( if ( ph ,  A ,  B )  =  A  ->  C  =  D )
ifsb.2  |-  ( if ( ph ,  A ,  B )  =  B  ->  C  =  E )
Assertion
Ref Expression
ifsb  |-  C  =  if ( ph ,  D ,  E )

Proof of Theorem ifsb
StepHypRef Expression
1 iftrue 3795 . . . 4  |-  ( ph  ->  if ( ph ,  A ,  B )  =  A )
2 ifsb.1 . . . 4  |-  ( if ( ph ,  A ,  B )  =  A  ->  C  =  D )
31, 2syl 16 . . 3  |-  ( ph  ->  C  =  D )
4 iftrue 3795 . . 3  |-  ( ph  ->  if ( ph ,  D ,  E )  =  D )
53, 4eqtr4d 2476 . 2  |-  ( ph  ->  C  =  if (
ph ,  D ,  E ) )
6 iffalse 3797 . . . 4  |-  ( -. 
ph  ->  if ( ph ,  A ,  B )  =  B )
7 ifsb.2 . . . 4  |-  ( if ( ph ,  A ,  B )  =  B  ->  C  =  E )
86, 7syl 16 . . 3  |-  ( -. 
ph  ->  C  =  E )
9 iffalse 3797 . . 3  |-  ( -. 
ph  ->  if ( ph ,  D ,  E )  =  E )
108, 9eqtr4d 2476 . 2  |-  ( -. 
ph  ->  C  =  if ( ph ,  D ,  E ) )
115, 10pm2.61i 164 1  |-  C  =  if ( ph ,  D ,  E )
Colors of variables: wff setvar class
Syntax hints:   -. wn 3    -> wi 4    = wceq 1369   ifcif 3789
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1591  ax-4 1602  ax-5 1670  ax-6 1708  ax-7 1728  ax-10 1775  ax-11 1780  ax-12 1792  ax-13 1943  ax-ext 2422
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-tru 1372  df-ex 1587  df-nf 1590  df-sb 1701  df-clab 2428  df-cleq 2434  df-clel 2437  df-if 3790
This theorem is referenced by:  fvif  5700  ovif  6166  ovif2  6167  xmulneg1  11230  ramcl  14088  matsc  18339  maducoeval2  18444  madugsum  18447  itg2const  21216  itg2monolem1  21226  iblmulc2  21306  itgmulc2lem1  21307  bddmulibl  21314  leibpi  22335  efrlim  22361  musumsum  22530  muinv  22531  dchrinvcl  22590  lgsneg  22656  lgsdilem  22659  dchrvmasumiflem2  22749  rpvmasum2  22759  padicabvcxp  22879  iffv  25902  itgaddnclem2  28448  itgmulc2nclem1  28455  ftc1anclem6  28469
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