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Theorem ifbieq2i 3758
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 3756 . . 3  |-  ( (
ph 
<->  ps )  ->  if ( ph ,  C ,  A )  =  if ( ps ,  C ,  A ) )
31, 2ax-mp 8 . 2  |-  if (
ph ,  C ,  A )  =  if ( ps ,  C ,  A )
4 ifbieq2i.2 . . 3  |-  A  =  B
5 ifeq2 3744 . . 3  |-  ( A  =  B  ->  if ( ps ,  C ,  A )  =  if ( ps ,  C ,  B ) )
64, 5ax-mp 8 . 2  |-  if ( ps ,  C ,  A )  =  if ( ps ,  C ,  B )
73, 6eqtri 2456 1  |-  if (
ph ,  C ,  A )  =  if ( ps ,  C ,  B )
Colors of variables: wff set class
Syntax hints:    <-> wb 177    = wceq 1652   ifcif 3739
This theorem is referenced by:  ifbieq12i  3760  gcdcom  13020  gcdass  13045  cdleme31sdnN  31184  cdlemefr44  31222  cdleme48fv  31296  cdlemeg49lebilem  31336  cdleme50eq  31338
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 2417
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 2423  df-cleq 2429  df-clel 2432  df-nfc 2561  df-rab 2714  df-v 2958  df-un 3325  df-if 3740
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