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Theorem ifeq2 3529
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 2722 . . 3  |-  ( A  =  B  ->  { x  e.  A  |  -.  ph }  =  { x  e.  B  |  -.  ph } )
21uneq2d 3281 . 2  |-  ( A  =  B  ->  ( { x  e.  C  |  ph }  u.  {
x  e.  A  |  -.  ph } )  =  ( { x  e.  C  |  ph }  u.  { x  e.  B  |  -.  ph } ) )
3 dfif6 3527 . 2  |-  if (
ph ,  C ,  A )  =  ( { x  e.  C  |  ph }  u.  {
x  e.  A  |  -.  ph } )
4 dfif6 3527 . 2  |-  if (
ph ,  C ,  B )  =  ( { x  e.  C  |  ph }  u.  {
x  e.  B  |  -.  ph } )
52, 3, 43eqtr4g 2228 1  |-  ( A  =  B  ->  if ( ph ,  C ,  A )  =  if ( ph ,  C ,  B ) )
Colors of variables: wff set class
Syntax hints:   -. wn 3    -> wi 4    = wceq 1348   {crab 2452    u. cun 3119   ifcif 3525
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-io 704  ax-5 1440  ax-7 1441  ax-gen 1442  ax-ie1 1486  ax-ie2 1487  ax-8 1497  ax-10 1498  ax-11 1499  ax-i12 1500  ax-bndl 1502  ax-4 1503  ax-17 1519  ax-i9 1523  ax-ial 1527  ax-i5r 1528  ax-ext 2152
This theorem depends on definitions:  df-bi 116  df-tru 1351  df-nf 1454  df-sb 1756  df-clab 2157  df-cleq 2163  df-clel 2166  df-nfc 2301  df-rab 2457  df-v 2732  df-un 3125  df-if 3526
This theorem is referenced by:  ifeq12  3541  ifeq2d  3543  ifbieq2i  3548  xrmaxiflemcom  11205
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