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Theorem ifeq1 3392
Description: Equality theorem for conditional operator. (Contributed by NM, 1-Sep-2004.) (Revised by Mario Carneiro, 8-Sep-2013.)
Assertion
Ref Expression
ifeq1  |-  ( A  =  B  ->  if ( ph ,  A ,  C )  =  if ( ph ,  B ,  C ) )

Proof of Theorem ifeq1
Dummy variable  x is distinct from all other variables.
StepHypRef Expression
1 rabeq 2611 . . 3  |-  ( A  =  B  ->  { x  e.  A  |  ph }  =  { x  e.  B  |  ph } )
21uneq1d 3151 . 2  |-  ( A  =  B  ->  ( { x  e.  A  |  ph }  u.  {
x  e.  C  |  -.  ph } )  =  ( { x  e.  B  |  ph }  u.  { x  e.  C  |  -.  ph } ) )
3 dfif6 3391 . 2  |-  if (
ph ,  A ,  C )  =  ( { x  e.  A  |  ph }  u.  {
x  e.  C  |  -.  ph } )
4 dfif6 3391 . 2  |-  if (
ph ,  B ,  C )  =  ( { x  e.  B  |  ph }  u.  {
x  e.  C  |  -.  ph } )
52, 3, 43eqtr4g 2145 1  |-  ( A  =  B  ->  if ( ph ,  A ,  C )  =  if ( ph ,  B ,  C ) )
Colors of variables: wff set class
Syntax hints:   -. wn 3    -> wi 4    = wceq 1289   {crab 2363    u. cun 2995   ifcif 3389
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106  ax-io 665  ax-5 1381  ax-7 1382  ax-gen 1383  ax-ie1 1427  ax-ie2 1428  ax-8 1440  ax-10 1441  ax-11 1442  ax-i12 1443  ax-bndl 1444  ax-4 1445  ax-17 1464  ax-i9 1468  ax-ial 1472  ax-i5r 1473  ax-ext 2070
This theorem depends on definitions:  df-bi 115  df-tru 1292  df-nf 1395  df-sb 1693  df-clab 2075  df-cleq 2081  df-clel 2084  df-nfc 2217  df-rab 2368  df-v 2621  df-un 3001  df-if 3390
This theorem is referenced by:  ifeq12  3403  ifeq1d  3404  ifbieq12i  3412  cbvsum  10713
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