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Theorem ifeq1d 3713
Description: Equality deduction for conditional operator. (Contributed by NM, 16-Feb-2005.)
Hypothesis
Ref Expression
ifeq1d.1  |-  ( ph  ->  A  =  B )
Assertion
Ref Expression
ifeq1d  |-  ( ph  ->  if ( ps ,  A ,  C )  =  if ( ps ,  B ,  C )
)

Proof of Theorem ifeq1d
StepHypRef Expression
1 ifeq1d.1 . 2  |-  ( ph  ->  A  =  B )
2 ifeq1 3703 . 2  |-  ( A  =  B  ->  if ( ps ,  A ,  C )  =  if ( ps ,  B ,  C ) )
31, 2syl 16 1  |-  ( ph  ->  if ( ps ,  A ,  C )  =  if ( ps ,  B ,  C )
)
Colors of variables: wff set class
Syntax hints:    -> wi 4    = wceq 1649   ifcif 3699
This theorem is referenced by:  ifeq12d  3715  ifeq1da  3724  riotabidva  6525  cantnflem1d  7600  cantnflem1  7601  isumless  12580  subgmulg  14913  gsumzsplit  15484  evlslem2  16523  cnmpt2pc  18906  pcoval2  18994  pcopt  19000  itgz  19625  iblss2  19650  itgss  19656  itgcn  19687  plyeq0lem  20082  dgrcolem2  20145  plydivlem4  20166  leibpi  20735  chtublem  20948  sumdchr  21009  bposlem6  21026  lgsval  21037  dchrvmasumiflem2  21149  padicabvcxp  21279  prodss  25226  dfrdg3  25367
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 1662  ax-8 1683  ax-6 1740  ax-7 1745  ax-11 1757  ax-12 1946  ax-ext 2385
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 2391  df-cleq 2397  df-clel 2400  df-nfc 2529  df-rab 2675  df-v 2918  df-un 3285  df-if 3700
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