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Theorem ifeq2 3570
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 2782 . . 3  |-  ( A  =  B  ->  { x  e.  A  |  -.  ph }  =  { x  e.  B  |  -.  ph } )
21uneq2d 3329 . 2  |-  ( A  =  B  ->  ( { x  e.  C  |  ph }  u.  {
x  e.  A  |  -.  ph } )  =  ( { x  e.  C  |  ph }  u.  { x  e.  B  |  -.  ph } ) )
3 dfif6 3568 . 2  |-  if (
ph ,  C ,  A )  =  ( { x  e.  C  |  ph }  u.  {
x  e.  A  |  -.  ph } )
4 dfif6 3568 . 2  |-  if (
ph ,  C ,  B )  =  ( { x  e.  C  |  ph }  u.  {
x  e.  B  |  -.  ph } )
52, 3, 43eqtr4g 2340 1  |-  ( A  =  B  ->  if ( ph ,  C ,  A )  =  if ( ph ,  C ,  B ) )
Colors of variables: wff set class
Syntax hints:   -. wn 3    -> wi 4    = wceq 1623   {crab 2547    u. cun 3150   ifcif 3565
This theorem is referenced by:  ifeq12  3578  ifeq2d  3580  ifbieq2i  3584  ifexg  3624  somincom  5080  prmorcht  20416  pclogsum  20454  cbvprodi  25312  hdmap1cbv  31993
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1533  ax-5 1544  ax-17 1603  ax-9 1635  ax-8 1643  ax-6 1703  ax-7 1708  ax-11 1715  ax-12 1866  ax-ext 2264
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-tru 1310  df-ex 1529  df-nf 1532  df-sb 1630  df-clab 2270  df-cleq 2276  df-clel 2279  df-nfc 2408  df-rab 2552  df-v 2790  df-un 3157  df-if 3566
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