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Theorem ifbieq2i 3465
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 3462 . . 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 3448 . . 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 2138 1  |-  if (
ph ,  C ,  A )  =  if ( ps ,  C ,  B )
Colors of variables: wff set class
Syntax hints:    <-> wb 104    = wceq 1316   ifcif 3444
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-in1 588  ax-in2 589  ax-io 683  ax-5 1408  ax-7 1409  ax-gen 1410  ax-ie1 1454  ax-ie2 1455  ax-8 1467  ax-10 1468  ax-11 1469  ax-i12 1470  ax-bndl 1471  ax-4 1472  ax-17 1491  ax-i9 1495  ax-ial 1499  ax-i5r 1500  ax-ext 2099
This theorem depends on definitions:  df-bi 116  df-tru 1319  df-nf 1422  df-sb 1721  df-clab 2104  df-cleq 2110  df-clel 2113  df-nfc 2247  df-rab 2402  df-v 2662  df-un 3045  df-if 3445
This theorem is referenced by:  ifbieq12i  3467  gcdcom  11589  gcdass  11630  lcmcom  11672  lcmass  11693
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