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Theorem ifeq1d 3697
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 3687 . 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 3683
This theorem is referenced by:  ifeq12d  3699  ifeq1da  3708  riotabidva  6503  cantnflem1d  7578  cantnflem1  7579  isumless  12553  subgmulg  14886  gsumzsplit  15457  evlslem2  16496  cnmpt2pc  18825  pcoval2  18913  pcopt  18919  itgz  19540  iblss2  19565  itgss  19571  itgcn  19602  plyeq0lem  19997  dgrcolem2  20060  plydivlem4  20081  leibpi  20650  chtublem  20863  sumdchr  20924  bposlem6  20941  lgsval  20952  dchrvmasumiflem2  21064  padicabvcxp  21194  prodss  25053  dfrdg3  25178
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 2369
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 2375  df-cleq 2381  df-clel 2384  df-nfc 2513  df-rab 2659  df-v 2902  df-un 3269  df-if 3684
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