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Theorem ifbieq2i 3930
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 3927 . . 3  |-  ( (
ph 
<->  ps )  ->  if ( ph ,  C ,  A )  =  if ( ps ,  C ,  A ) )
31, 2ax-mp 5 . 2  |-  if (
ph ,  C ,  A )  =  if ( ps ,  C ,  A )
4 ifbieq2i.2 . . 3  |-  A  =  B
5 ifeq2 3911 . . 3  |-  ( A  =  B  ->  if ( ps ,  C ,  A )  =  if ( ps ,  C ,  B ) )
64, 5ax-mp 5 . 2  |-  if ( ps ,  C ,  A )  =  if ( ps ,  C ,  B )
73, 6eqtri 2449 1  |-  if (
ph ,  C ,  A )  =  if ( ps ,  C ,  B )
Colors of variables: wff setvar class
Syntax hints:    <-> wb 187    = wceq 1437   ifcif 3906
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1665  ax-4 1678  ax-5 1748  ax-6 1794  ax-7 1838  ax-10 1886  ax-11 1891  ax-12 1904  ax-13 2052  ax-ext 2398
This theorem depends on definitions:  df-bi 188  df-or 371  df-an 372  df-tru 1440  df-ex 1660  df-nf 1664  df-sb 1787  df-clab 2406  df-cleq 2412  df-clel 2415  df-nfc 2570  df-rab 2782  df-v 3080  df-un 3438  df-if 3907
This theorem is referenced by:  ifbieq12i  3932  gcdcom  14462  gcdass  14491  lcmcom  14535  lcmass  14557  bj-xpimasn  31498  cdleme31sdnN  33867  cdlemefr44  33905  cdleme48fv  33979  cdlemeg49lebilem  34019  cdleme50eq  34021
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