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Theorem ifbieq12i 3420
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 3400 . . 3  |-  ( A  =  C  ->  if ( ph ,  A ,  B )  =  if ( ph ,  C ,  B ) )
31, 2ax-mp 7 . 2  |-  if (
ph ,  A ,  B )  =  if ( ph ,  C ,  B )
4 ifbieq12i.1 . . 3  |-  ( ph  <->  ps )
5 ifbieq12i.3 . . 3  |-  B  =  D
64, 5ifbieq2i 3418 . 2  |-  if (
ph ,  C ,  B )  =  if ( ps ,  C ,  D )
73, 6eqtri 2109 1  |-  if (
ph ,  A ,  B )  =  if ( ps ,  C ,  D )
Colors of variables: wff set class
Syntax hints:    <-> wb 104    = wceq 1290   ifcif 3397
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-in1 580  ax-in2 581  ax-io 666  ax-5 1382  ax-7 1383  ax-gen 1384  ax-ie1 1428  ax-ie2 1429  ax-8 1441  ax-10 1442  ax-11 1443  ax-i12 1444  ax-bndl 1445  ax-4 1446  ax-17 1465  ax-i9 1469  ax-ial 1473  ax-i5r 1474  ax-ext 2071
This theorem depends on definitions:  df-bi 116  df-tru 1293  df-nf 1396  df-sb 1694  df-clab 2076  df-cleq 2082  df-clel 2085  df-nfc 2218  df-rab 2369  df-v 2622  df-un 3004  df-if 3398
This theorem is referenced by: (None)
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