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Theorem ifeq1d 3745
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 3735 . 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 1652   ifcif 3731
This theorem is referenced by:  ifeq12d  3747  ifeq1da  3756  riotabidva  6558  cantnflem1d  7634  cantnflem1  7635  isumless  12615  subgmulg  14948  gsumzsplit  15519  evlslem2  16558  cnmpt2pc  18943  pcoval2  19031  pcopt  19037  itgz  19662  iblss2  19687  itgss  19693  itgcn  19724  plyeq0lem  20119  dgrcolem2  20182  plydivlem4  20203  leibpi  20772  chtublem  20985  sumdchr  21046  bposlem6  21063  lgsval  21074  dchrvmasumiflem2  21186  padicabvcxp  21316  prodss  25263  dfrdg3  25412
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1555  ax-5 1566  ax-17 1626  ax-9 1666  ax-8 1687  ax-6 1744  ax-7 1749  ax-11 1761  ax-12 1950  ax-ext 2416
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-tru 1328  df-ex 1551  df-nf 1554  df-sb 1659  df-clab 2422  df-cleq 2428  df-clel 2431  df-nfc 2560  df-rab 2706  df-v 2950  df-un 3317  df-if 3732
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