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Theorem ifbieq12i 3752
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 3735 . . 3  |-  ( A  =  C  ->  if ( ph ,  A ,  B )  =  if ( ph ,  C ,  B ) )
31, 2ax-mp 8 . 2  |-  if (
ph ,  A ,  B )  =  if ( ph ,  C ,  B )
4 ifbieq12i.1 . . 3  |-  ( ph  <->  ps )
5 ifbieq12i.3 . . 3  |-  B  =  D
64, 5ifbieq2i 3750 . 2  |-  if (
ph ,  C ,  B )  =  if ( ps ,  C ,  D )
73, 6eqtri 2455 1  |-  if (
ph ,  A ,  B )  =  if ( ps ,  C ,  D )
Colors of variables: wff set class
Syntax hints:    <-> wb 177    = wceq 1652   ifcif 3731
This theorem is referenced by:  cbvriota  6552  cbvditg  19733
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 2416
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 2422  df-cleq 2428  df-clel 2431  df-nfc 2560  df-rab 2706  df-v 2950  df-un 3317  df-if 3732
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