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Theorem ifbieq1d 3350
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 3349 . 2  |-  ( ph  ->  if ( ps ,  A ,  C )  =  if ( ch ,  A ,  C )
)
3 ifbieq1d.2 . . 3  |-  ( ph  ->  A  =  B )
43ifeq1d 3345 . 2  |-  ( ph  ->  if ( ch ,  A ,  C )  =  if ( ch ,  B ,  C )
)
52, 4eqtrd 2072 1  |-  ( ph  ->  if ( ps ,  A ,  C )  =  if ( ch ,  B ,  C )
)
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 98    = wceq 1243   ifcif 3331
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 99  ax-ia2 100  ax-ia3 101  ax-in1 544  ax-in2 545  ax-io 630  ax-5 1336  ax-7 1337  ax-gen 1338  ax-ie1 1382  ax-ie2 1383  ax-8 1395  ax-10 1396  ax-11 1397  ax-i12 1398  ax-bndl 1399  ax-4 1400  ax-17 1419  ax-i9 1423  ax-ial 1427  ax-i5r 1428  ax-ext 2022
This theorem depends on definitions:  df-bi 110  df-tru 1246  df-nf 1350  df-sb 1646  df-clab 2027  df-cleq 2033  df-clel 2036  df-nfc 2167  df-rab 2315  df-v 2559  df-un 2922  df-if 3332
This theorem is referenced by: (None)
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