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Theorem ifeq2 3583
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 2795 . . 3  |-  ( A  =  B  ->  { x  e.  A  |  -.  ph }  =  { x  e.  B  |  -.  ph } )
21uneq2d 3342 . 2  |-  ( A  =  B  ->  ( { x  e.  C  |  ph }  u.  {
x  e.  A  |  -.  ph } )  =  ( { x  e.  C  |  ph }  u.  { x  e.  B  |  -.  ph } ) )
3 dfif6 3581 . 2  |-  if (
ph ,  C ,  A )  =  ( { x  e.  C  |  ph }  u.  {
x  e.  A  |  -.  ph } )
4 dfif6 3581 . 2  |-  if (
ph ,  C ,  B )  =  ( { x  e.  C  |  ph }  u.  {
x  e.  B  |  -.  ph } )
52, 3, 43eqtr4g 2353 1  |-  ( A  =  B  ->  if ( ph ,  C ,  A )  =  if ( ph ,  C ,  B ) )
Colors of variables: wff set class
Syntax hints:   -. wn 3    -> wi 4    = wceq 1632   {crab 2560    u. cun 3163   ifcif 3578
This theorem is referenced by:  ifeq12  3591  ifeq2d  3593  ifbieq2i  3597  ifexg  3637  somincom  5096  prmorcht  20432  pclogsum  20470  cbvprodi  25415  hdmap1cbv  32615
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1536  ax-5 1547  ax-17 1606  ax-9 1644  ax-8 1661  ax-6 1715  ax-7 1720  ax-11 1727  ax-12 1878  ax-ext 2277
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-tru 1310  df-ex 1532  df-nf 1535  df-sb 1639  df-clab 2283  df-cleq 2289  df-clel 2292  df-nfc 2421  df-rab 2565  df-v 2803  df-un 3170  df-if 3579
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