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Theorem ifeq2da 3701
Description: Conditional equality. (Contributed by Jeff Madsen, 2-Sep-2009.)
Hypothesis
Ref Expression
ifeq2da.1  |-  ( (
ph  /\  -.  ps )  ->  A  =  B )
Assertion
Ref Expression
ifeq2da  |-  ( ph  ->  if ( ps ,  C ,  A )  =  if ( ps ,  C ,  B )
)

Proof of Theorem ifeq2da
StepHypRef Expression
1 iftrue 3681 . . . 4  |-  ( ps 
->  if ( ps ,  C ,  A )  =  C )
2 iftrue 3681 . . . 4  |-  ( ps 
->  if ( ps ,  C ,  B )  =  C )
31, 2eqtr4d 2415 . . 3  |-  ( ps 
->  if ( ps ,  C ,  A )  =  if ( ps ,  C ,  B )
)
43adantl 453 . 2  |-  ( (
ph  /\  ps )  ->  if ( ps ,  C ,  A )  =  if ( ps ,  C ,  B )
)
5 ifeq2da.1 . . 3  |-  ( (
ph  /\  -.  ps )  ->  A  =  B )
65ifeq2d 3690 . 2  |-  ( (
ph  /\  -.  ps )  ->  if ( ps ,  C ,  A )  =  if ( ps ,  C ,  B )
)
74, 6pm2.61dan 767 1  |-  ( ph  ->  if ( ps ,  C ,  A )  =  if ( ps ,  C ,  B )
)
Colors of variables: wff set class
Syntax hints:   -. wn 3    -> wi 4    /\ wa 359    = wceq 1649   ifcif 3675
This theorem is referenced by:  dfac12lem1  7949  ttukeylem3  8317  xmulcom  10770  xmulneg1  10773  subgmulg  14878  copco  18907  pcopt2  18912  limcdif  19623  limcmpt  19630  limcres  19633  limccnp  19638  radcnv0  20192  leibpi  20642  efrlim  20668  dchrvmasumiflem2  21056  rpvmasum2  21066  padicabvf  21185  padicabvcxp  21186  itg2addnclem  25950
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1552  ax-5 1563  ax-17 1623  ax-9 1661  ax-8 1682  ax-6 1736  ax-7 1741  ax-11 1753  ax-12 1939  ax-ext 2361
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-tru 1325  df-ex 1548  df-nf 1551  df-sb 1656  df-clab 2367  df-cleq 2373  df-clel 2376  df-nfc 2505  df-rab 2651  df-v 2894  df-un 3261  df-if 3676
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