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Theorem ifeq2 3687
Description: Equality theorem for conditional operator. (Contributed by NM, 1-Sep-2004.) (Revised by Mario Carneiro, 8-Sep-2013.)
Assertion
Ref Expression
ifeq2  |-  ( A  =  B  ->  if ( ph ,  C ,  A )  =  if ( ph ,  C ,  B ) )

Proof of Theorem ifeq2
Dummy variable  x is distinct from all other variables.
StepHypRef Expression
1 rabeq 2893 . . 3  |-  ( A  =  B  ->  { x  e.  A  |  -.  ph }  =  { x  e.  B  |  -.  ph } )
21uneq2d 3444 . 2  |-  ( A  =  B  ->  ( { x  e.  C  |  ph }  u.  {
x  e.  A  |  -.  ph } )  =  ( { x  e.  C  |  ph }  u.  { x  e.  B  |  -.  ph } ) )
3 dfif6 3685 . 2  |-  if (
ph ,  C ,  A )  =  ( { x  e.  C  |  ph }  u.  {
x  e.  A  |  -.  ph } )
4 dfif6 3685 . 2  |-  if (
ph ,  C ,  B )  =  ( { x  e.  C  |  ph }  u.  {
x  e.  B  |  -.  ph } )
52, 3, 43eqtr4g 2444 1  |-  ( A  =  B  ->  if ( ph ,  C ,  A )  =  if ( ph ,  C ,  B ) )
Colors of variables: wff set class
Syntax hints:   -. wn 3    -> wi 4    = wceq 1649   {crab 2653    u. cun 3261   ifcif 3682
This theorem is referenced by:  ifeq12  3695  ifeq2d  3697  ifbieq2i  3701  ifexg  3741  somincom  5211  prmorcht  20828  pclogsum  20866  hdmap1cbv  31918
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1552  ax-5 1563  ax-17 1623  ax-9 1661  ax-8 1682  ax-6 1736  ax-7 1741  ax-11 1753  ax-12 1939  ax-ext 2368
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-tru 1325  df-ex 1548  df-nf 1551  df-sb 1656  df-clab 2374  df-cleq 2380  df-clel 2383  df-nfc 2512  df-rab 2658  df-v 2901  df-un 3268  df-if 3683
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