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Theorem ifbieq2i 3412
Description: Equivalence/equality inference for conditional operators. (Contributed by Paul Chapman, 22-Jun-2011.)
Hypotheses
Ref Expression
ifbieq2i.1  |-  ( ph  <->  ps )
ifbieq2i.2  |-  A  =  B
Assertion
Ref Expression
ifbieq2i  |-  if (
ph ,  C ,  A )  =  if ( ps ,  C ,  B )

Proof of Theorem ifbieq2i
StepHypRef Expression
1 ifbieq2i.1 . . 3  |-  ( ph  <->  ps )
2 ifbi 3409 . . 3  |-  ( (
ph 
<->  ps )  ->  if ( ph ,  C ,  A )  =  if ( ps ,  C ,  A ) )
31, 2ax-mp 7 . 2  |-  if (
ph ,  C ,  A )  =  if ( ps ,  C ,  A )
4 ifbieq2i.2 . . 3  |-  A  =  B
5 ifeq2 3395 . . 3  |-  ( A  =  B  ->  if ( ps ,  C ,  A )  =  if ( ps ,  C ,  B ) )
64, 5ax-mp 7 . 2  |-  if ( ps ,  C ,  A )  =  if ( ps ,  C ,  B )
73, 6eqtri 2108 1  |-  if (
ph ,  C ,  A )  =  if ( ps ,  C ,  B )
Colors of variables: wff set class
Syntax hints:    <-> wb 103    = wceq 1289   ifcif 3391
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 579  ax-in2 580  ax-io 665  ax-5 1381  ax-7 1382  ax-gen 1383  ax-ie1 1427  ax-ie2 1428  ax-8 1440  ax-10 1441  ax-11 1442  ax-i12 1443  ax-bndl 1444  ax-4 1445  ax-17 1464  ax-i9 1468  ax-ial 1472  ax-i5r 1473  ax-ext 2070
This theorem depends on definitions:  df-bi 115  df-tru 1292  df-nf 1395  df-sb 1693  df-clab 2075  df-cleq 2081  df-clel 2084  df-nfc 2217  df-rab 2368  df-v 2621  df-un 3003  df-if 3392
This theorem is referenced by:  ifbieq12i  3414  gcdcom  11230  gcdass  11269  lcmcom  11311  lcmass  11332
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