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Theorem ifbieq1d 3464
Description: Equivalence/equality deduction for conditional operators. (Contributed by JJ, 25-Sep-2018.)
Hypotheses
Ref Expression
ifbieq1d.1  |-  ( ph  ->  ( ps  <->  ch )
)
ifbieq1d.2  |-  ( ph  ->  A  =  B )
Assertion
Ref Expression
ifbieq1d  |-  ( ph  ->  if ( ps ,  A ,  C )  =  if ( ch ,  B ,  C )
)

Proof of Theorem ifbieq1d
StepHypRef Expression
1 ifbieq1d.1 . . 3  |-  ( ph  ->  ( ps  <->  ch )
)
21ifbid 3463 . 2  |-  ( ph  ->  if ( ps ,  A ,  C )  =  if ( ch ,  A ,  C )
)
3 ifbieq1d.2 . . 3  |-  ( ph  ->  A  =  B )
43ifeq1d 3459 . 2  |-  ( ph  ->  if ( ch ,  A ,  C )  =  if ( ch ,  B ,  C )
)
52, 4eqtrd 2150 1  |-  ( ph  ->  if ( ps ,  A ,  C )  =  if ( ch ,  B ,  C )
)
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 104    = wceq 1316   ifcif 3444
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-in1 588  ax-in2 589  ax-io 683  ax-5 1408  ax-7 1409  ax-gen 1410  ax-ie1 1454  ax-ie2 1455  ax-8 1467  ax-10 1468  ax-11 1469  ax-i12 1470  ax-bndl 1471  ax-4 1472  ax-17 1491  ax-i9 1495  ax-ial 1499  ax-i5r 1500  ax-ext 2099
This theorem depends on definitions:  df-bi 116  df-tru 1319  df-nf 1422  df-sb 1721  df-clab 2104  df-cleq 2110  df-clel 2113  df-nfc 2247  df-rab 2402  df-v 2662  df-un 3045  df-if 3445
This theorem is referenced by:  ctssdclemn0  6963  ctssdc  6966  enumctlemm  6967  iseqf1olemfvp  10225  seq3f1olemqsum  10228  seq3f1oleml  10231  seq3f1o  10232  bcval  10450  sumrbdclem  11100  summodclem3  11104  summodclem2a  11105  summodc  11107  zsumdc  11108  fsum3  11111  isumss  11115  isumss2  11117  fsum3cvg2  11118  fsum3ser  11121  fsumcl2lem  11122  fsumadd  11130  sumsnf  11133  fsummulc2  11172  isumlessdc  11220
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