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Theorem ifbieq2d 3395
Description: Equivalence/equality deduction for conditional operators. (Contributed by Paul Chapman, 22-Jun-2011.)
Hypotheses
Ref Expression
ifbieq2d.1  |-  ( ph  ->  ( ps  <->  ch )
)
ifbieq2d.2  |-  ( ph  ->  A  =  B )
Assertion
Ref Expression
ifbieq2d  |-  ( ph  ->  if ( ps ,  C ,  A )  =  if ( ch ,  C ,  B )
)

Proof of Theorem ifbieq2d
StepHypRef Expression
1 ifbieq2d.1 . . 3  |-  ( ph  ->  ( ps  <->  ch )
)
21ifbid 3392 . 2  |-  ( ph  ->  if ( ps ,  C ,  A )  =  if ( ch ,  C ,  A )
)
3 ifbieq2d.2 . . 3  |-  ( ph  ->  A  =  B )
43ifeq2d 3389 . 2  |-  ( ph  ->  if ( ch ,  C ,  A )  =  if ( ch ,  C ,  B )
)
52, 4eqtrd 2115 1  |-  ( ph  ->  if ( ps ,  C ,  A )  =  if ( ch ,  C ,  B )
)
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 103    = wceq 1285   ifcif 3373
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106  ax-in1 577  ax-in2 578  ax-io 663  ax-5 1377  ax-7 1378  ax-gen 1379  ax-ie1 1423  ax-ie2 1424  ax-8 1436  ax-10 1437  ax-11 1438  ax-i12 1439  ax-bndl 1440  ax-4 1441  ax-17 1460  ax-i9 1464  ax-ial 1468  ax-i5r 1469  ax-ext 2065
This theorem depends on definitions:  df-bi 115  df-tru 1288  df-nf 1391  df-sb 1688  df-clab 2070  df-cleq 2076  df-clel 2079  df-nfc 2212  df-rab 2362  df-v 2614  df-un 2988  df-if 3374
This theorem is referenced by:  xnegeq  9184  expival  9794  gcdval  10731  gcdass  10784  lcmval  10825  lcmass  10847
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