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Theorem ifeq1 3688
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 2895 . . 3  |-  ( A  =  B  ->  { x  e.  A  |  ph }  =  { x  e.  B  |  ph } )
21uneq1d 3445 . 2  |-  ( A  =  B  ->  ( { x  e.  A  |  ph }  u.  {
x  e.  C  |  -.  ph } )  =  ( { x  e.  B  |  ph }  u.  { x  e.  C  |  -.  ph } ) )
3 dfif6 3687 . 2  |-  if (
ph ,  A ,  C )  =  ( { x  e.  A  |  ph }  u.  {
x  e.  C  |  -.  ph } )
4 dfif6 3687 . 2  |-  if (
ph ,  B ,  C )  =  ( { x  e.  B  |  ph }  u.  {
x  e.  C  |  -.  ph } )
52, 3, 43eqtr4g 2446 1  |-  ( A  =  B  ->  if ( ph ,  A ,  C )  =  if ( ph ,  B ,  C ) )
Colors of variables: wff set class
Syntax hints:   -. wn 3    -> wi 4    = wceq 1649   {crab 2655    u. cun 3263   ifcif 3684
This theorem is referenced by:  ifeq12  3697  ifeq1d  3698  ifbieq12i  3705  ifexg  3743  rdgeq2  6608  dfoi  7415  wemaplem2  7451  cantnflem1  7580  sumeq2w  12415  sumeq2ii  12416  mplcoe3  16458  ellimc  19629  ply1nzb  19914  dchrvmasumiflem1  21064  prodeq2w  25019  prodeq2ii  25020  dfrdg2  25178  dfafv2  27667
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1552  ax-5 1563  ax-17 1623  ax-9 1661  ax-8 1682  ax-6 1736  ax-7 1741  ax-11 1753  ax-12 1939  ax-ext 2370
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-tru 1325  df-ex 1548  df-nf 1551  df-sb 1656  df-clab 2376  df-cleq 2382  df-clel 2385  df-nfc 2514  df-rab 2660  df-v 2903  df-un 3270  df-if 3685
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