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Theorem ifeq2da 3931
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 3908 . . . 4  |-  ( ps 
->  if ( ps ,  C ,  A )  =  C )
2 iftrue 3908 . . . 4  |-  ( ps 
->  if ( ps ,  C ,  B )  =  C )
31, 2eqtr4d 2498 . . 3  |-  ( ps 
->  if ( ps ,  C ,  A )  =  if ( ps ,  C ,  B )
)
43adantl 466 . 2  |-  ( (
ph  /\  ps )  ->  if ( ps ,  C ,  A )  =  if ( ps ,  C ,  B )
)
5 ifeq2da.1 . . 3  |-  ( (
ph  /\  -.  ps )  ->  A  =  B )
65ifeq2d 3919 . 2  |-  ( (
ph  /\  -.  ps )  ->  if ( ps ,  C ,  A )  =  if ( ps ,  C ,  B )
)
74, 6pm2.61dan 789 1  |-  ( ph  ->  if ( ps ,  C ,  A )  =  if ( ps ,  C ,  B )
)
Colors of variables: wff setvar class
Syntax hints:   -. wn 3    -> wi 4    /\ wa 369    = wceq 1370   ifcif 3902
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1592  ax-4 1603  ax-5 1671  ax-6 1710  ax-7 1730  ax-10 1777  ax-11 1782  ax-12 1794  ax-13 1955  ax-ext 2432
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-tru 1373  df-ex 1588  df-nf 1591  df-sb 1703  df-clab 2440  df-cleq 2446  df-clel 2449  df-nfc 2604  df-rab 2808  df-v 3080  df-un 3444  df-if 3903
This theorem is referenced by:  dfac12lem1  8426  ttukeylem3  8794  xmulcom  11343  xmulneg1  11346  subgmulg  15817  1marepvmarrepid  18516  copco  20725  pcopt2  20730  limcdif  21487  limcmpt  21494  limcres  21497  limccnp  21502  radcnv0  22017  leibpi  22473  efrlim  22499  dchrvmasumiflem2  22887  rpvmasum2  22897  padicabvf  23016  padicabvcxp  23017  itg2addnclem  28611
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