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Theorem ifbieq12i 3571
Description: Equivalence deduction for conditional operators. (Contributed by NM, 18-Mar-2013.)
Hypotheses
Ref Expression
ifbieq12i.1  |-  ( ph  <->  ps )
ifbieq12i.2  |-  A  =  C
ifbieq12i.3  |-  B  =  D
Assertion
Ref Expression
ifbieq12i  |-  if (
ph ,  A ,  B )  =  if ( ps ,  C ,  D )

Proof of Theorem ifbieq12i
StepHypRef Expression
1 ifbieq12i.2 . . 3  |-  A  =  C
2 ifeq1 3549 . . 3  |-  ( A  =  C  ->  if ( ph ,  A ,  B )  =  if ( ph ,  C ,  B ) )
31, 2ax-mp 5 . 2  |-  if (
ph ,  A ,  B )  =  if ( ph ,  C ,  B )
4 ifbieq12i.1 . . 3  |-  ( ph  <->  ps )
5 ifbieq12i.3 . . 3  |-  B  =  D
64, 5ifbieq2i 3569 . 2  |-  if (
ph ,  C ,  B )  =  if ( ps ,  C ,  D )
73, 6eqtri 2208 1  |-  if (
ph ,  A ,  B )  =  if ( ps ,  C ,  D )
Colors of variables: wff set class
Syntax hints:    <-> wb 105    = wceq 1363   ifcif 3546
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-in1 615  ax-in2 616  ax-io 710  ax-5 1457  ax-7 1458  ax-gen 1459  ax-ie1 1503  ax-ie2 1504  ax-8 1514  ax-10 1515  ax-11 1516  ax-i12 1517  ax-bndl 1519  ax-4 1520  ax-17 1536  ax-i9 1540  ax-ial 1544  ax-i5r 1545  ax-ext 2169
This theorem depends on definitions:  df-bi 117  df-tru 1366  df-nf 1471  df-sb 1773  df-clab 2174  df-cleq 2180  df-clel 2183  df-nfc 2318  df-rab 2474  df-v 2751  df-un 3145  df-if 3547
This theorem is referenced by: (None)
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