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Theorem ifeq1 3474
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
StepHypRef Expression
1 rabeq 2721 . . 3  |-  ( A  =  B  ->  { x  e.  A  |  ph }  =  { x  e.  B  |  ph } )
21uneq1d 3238 . 2  |-  ( A  =  B  ->  ( { x  e.  A  |  ph }  u.  {
x  e.  C  |  -.  ph } )  =  ( { x  e.  B  |  ph }  u.  { x  e.  C  |  -.  ph } ) )
3 dfif6 3473 . 2  |-  if (
ph ,  A ,  C )  =  ( { x  e.  A  |  ph }  u.  {
x  e.  C  |  -.  ph } )
4 dfif6 3473 . 2  |-  if (
ph ,  B ,  C )  =  ( { x  e.  B  |  ph }  u.  {
x  e.  C  |  -.  ph } )
52, 3, 43eqtr4g 2310 1  |-  ( A  =  B  ->  if ( ph ,  A ,  C )  =  if ( ph ,  B ,  C ) )
Colors of variables: wff set class
Syntax hints:   -. wn 5    -> wi 6    = wceq 1619   {crab 2512    u. cun 3076   ifcif 3470
This theorem is referenced by:  ifeq12  3483  ifeq1d  3484  ifbieq12i  3491  ifexg  3529  rdgeq2  6311  dfoi  7110  wemaplem2  7146  cantnflem1  7275  sumeq2w  12042  sumeq2ii  12043  mplcoe3  16042  ellimc  19055  ply1nzb  19340  dchrvmasumiflem1  20482  dfrdg2  23320
This theorem was proved from axioms:  ax-1 7  ax-2 8  ax-3 9  ax-mp 10  ax-5 1533  ax-6 1534  ax-7 1535  ax-gen 1536  ax-8 1623  ax-11 1624  ax-17 1628  ax-12o 1664  ax-10 1678  ax-9 1684  ax-4 1692  ax-16 1926  ax-ext 2234
This theorem depends on definitions:  df-bi 179  df-or 361  df-an 362  df-tru 1315  df-ex 1538  df-nf 1540  df-sb 1883  df-clab 2240  df-cleq 2246  df-clel 2249  df-nfc 2374  df-rab 2516  df-v 2729  df-un 3083  df-if 3471
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