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Theorem ifsb 3574
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 3571 . . . 4  |-  ( ph  ->  if ( ph ,  A ,  B )  =  A )
2 ifsb.1 . . . 4  |-  ( if ( ph ,  A ,  B )  =  A  ->  C  =  D )
31, 2syl 15 . . 3  |-  ( ph  ->  C  =  D )
4 iftrue 3571 . . 3  |-  ( ph  ->  if ( ph ,  D ,  E )  =  D )
53, 4eqtr4d 2318 . 2  |-  ( ph  ->  C  =  if (
ph ,  D ,  E ) )
6 iffalse 3572 . . . 4  |-  ( -. 
ph  ->  if ( ph ,  A ,  B )  =  B )
7 ifsb.2 . . . 4  |-  ( if ( ph ,  A ,  B )  =  B  ->  C  =  E )
86, 7syl 15 . . 3  |-  ( -. 
ph  ->  C  =  E )
9 iffalse 3572 . . 3  |-  ( -. 
ph  ->  if ( ph ,  D ,  E )  =  E )
108, 9eqtr4d 2318 . 2  |-  ( -. 
ph  ->  C  =  if ( ph ,  D ,  E ) )
115, 10pm2.61i 156 1  |-  C  =  if ( ph ,  D ,  E )
Colors of variables: wff set class
Syntax hints:   -. wn 3    -> wi 4    = wceq 1623   ifcif 3565
This theorem is referenced by:  fvif  5540  xmulneg1  10589  ramcl  13076  itg2const  19095  itg2monolem1  19105  iblmulc2  19185  itgmulc2lem1  19186  bddmulibl  19193  leibpi  20238  efrlim  20264  musumsum  20432  muinv  20433  dchrinvcl  20492  lgsneg  20558  lgsdilem  20561  dchrvmasumiflem2  20651  rpvmasum2  20661  padicabvcxp  20781
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-tru 1310  df-ex 1529  df-nf 1532  df-sb 1630  df-clab 2270  df-cleq 2276  df-clel 2279  df-if 3566
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