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Theorem ifeq2 3736
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 2942 . . 3  |-  ( A  =  B  ->  { x  e.  A  |  -.  ph }  =  { x  e.  B  |  -.  ph } )
21uneq2d 3493 . 2  |-  ( A  =  B  ->  ( { x  e.  C  |  ph }  u.  {
x  e.  A  |  -.  ph } )  =  ( { x  e.  C  |  ph }  u.  { x  e.  B  |  -.  ph } ) )
3 dfif6 3734 . 2  |-  if (
ph ,  C ,  A )  =  ( { x  e.  C  |  ph }  u.  {
x  e.  A  |  -.  ph } )
4 dfif6 3734 . 2  |-  if (
ph ,  C ,  B )  =  ( { x  e.  C  |  ph }  u.  {
x  e.  B  |  -.  ph } )
52, 3, 43eqtr4g 2492 1  |-  ( A  =  B  ->  if ( ph ,  C ,  A )  =  if ( ph ,  C ,  B ) )
Colors of variables: wff set class
Syntax hints:   -. wn 3    -> wi 4    = wceq 1652   {crab 2701    u. cun 3310   ifcif 3731
This theorem is referenced by:  ifeq12  3744  ifeq2d  3746  ifbieq2i  3750  ifexg  3790  somincom  5263  prmorcht  20951  pclogsum  20989  hdmap1cbv  32502
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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