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Theorem ifeq1 3948
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 3103 . . 3  |-  ( A  =  B  ->  { x  e.  A  |  ph }  =  { x  e.  B  |  ph } )
21uneq1d 3653 . 2  |-  ( A  =  B  ->  ( { x  e.  A  |  ph }  u.  {
x  e.  C  |  -.  ph } )  =  ( { x  e.  B  |  ph }  u.  { x  e.  C  |  -.  ph } ) )
3 dfif6 3947 . 2  |-  if (
ph ,  A ,  C )  =  ( { x  e.  A  |  ph }  u.  {
x  e.  C  |  -.  ph } )
4 dfif6 3947 . 2  |-  if (
ph ,  B ,  C )  =  ( { x  e.  B  |  ph }  u.  {
x  e.  C  |  -.  ph } )
52, 3, 43eqtr4g 2523 1  |-  ( A  =  B  ->  if ( ph ,  A ,  C )  =  if ( ph ,  B ,  C ) )
Colors of variables: wff setvar class
Syntax hints:   -. wn 3    -> wi 4    = wceq 1395   {crab 2811    u. cun 3469   ifcif 3944
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1619  ax-4 1632  ax-5 1705  ax-6 1748  ax-7 1791  ax-10 1838  ax-11 1843  ax-12 1855  ax-13 2000  ax-ext 2435
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-tru 1398  df-ex 1614  df-nf 1618  df-sb 1741  df-clab 2443  df-cleq 2449  df-clel 2452  df-nfc 2607  df-rab 2816  df-v 3111  df-un 3476  df-if 3945
This theorem is referenced by:  ifeq12  3961  ifeq1d  3962  ifbieq12i  3970  ifexg  4014  rdgeq2  7096  dfoi  7954  wemaplem2  7990  cantnflem1  8125  cantnflem1OLD  8148  prodeq2w  13730  prodeq2ii  13731  mgm2nsgrplem2  16163  mgm2nsgrplem3  16164  mplcoe3  18254  mplcoe3OLD  18255  marrepval0  19189  ellimc  22402  ply1nzb  22648  dchrvmasumiflem1  23811  signspval  28684  dfrdg2  29402  dfafv2  32378
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