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Theorem ifeq1 3703
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 2910 . . 3  |-  ( A  =  B  ->  { x  e.  A  |  ph }  =  { x  e.  B  |  ph } )
21uneq1d 3460 . 2  |-  ( A  =  B  ->  ( { x  e.  A  |  ph }  u.  {
x  e.  C  |  -.  ph } )  =  ( { x  e.  B  |  ph }  u.  { x  e.  C  |  -.  ph } ) )
3 dfif6 3702 . 2  |-  if (
ph ,  A ,  C )  =  ( { x  e.  A  |  ph }  u.  {
x  e.  C  |  -.  ph } )
4 dfif6 3702 . 2  |-  if (
ph ,  B ,  C )  =  ( { x  e.  B  |  ph }  u.  {
x  e.  C  |  -.  ph } )
52, 3, 43eqtr4g 2461 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 2670    u. cun 3278   ifcif 3699
This theorem is referenced by:  ifeq12  3712  ifeq1d  3713  ifbieq12i  3720  ifexg  3758  rdgeq2  6629  dfoi  7436  wemaplem2  7472  cantnflem1  7601  sumeq2w  12441  sumeq2ii  12442  mplcoe3  16484  ellimc  19713  ply1nzb  19998  dchrvmasumiflem1  21148  prodeq2w  25191  prodeq2ii  25192  dfrdg2  25366  dfafv2  27863
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 1662  ax-8 1683  ax-6 1740  ax-7 1745  ax-11 1757  ax-12 1946  ax-ext 2385
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 2391  df-cleq 2397  df-clel 2400  df-nfc 2529  df-rab 2675  df-v 2918  df-un 3285  df-if 3700
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