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Theorem ifexg 3984
Description: Conditional operator existence. (Contributed by NM, 21-Mar-2011.)
Assertion
Ref Expression
ifexg  |-  ( ( A  e.  V  /\  B  e.  W )  ->  if ( ph ,  A ,  B )  e.  _V )

Proof of Theorem ifexg
Dummy variables  x  y are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 ifeq1 3919 . . 3  |-  ( x  =  A  ->  if ( ph ,  x ,  y )  =  if ( ph ,  A ,  y ) )
21eleq1d 2498 . 2  |-  ( x  =  A  ->  ( if ( ph ,  x ,  y )  e. 
_V 
<->  if ( ph ,  A ,  y )  e.  _V ) )
3 ifeq2 3920 . . 3  |-  ( y  =  B  ->  if ( ph ,  A , 
y )  =  if ( ph ,  A ,  B ) )
43eleq1d 2498 . 2  |-  ( y  =  B  ->  ( if ( ph ,  A ,  y )  e. 
_V 
<->  if ( ph ,  A ,  B )  e.  _V ) )
5 vex 3090 . . 3  |-  x  e. 
_V
6 vex 3090 . . 3  |-  y  e. 
_V
75, 6ifex 3983 . 2  |-  if (
ph ,  x ,  y )  e.  _V
82, 4, 7vtocl2g 3149 1  |-  ( ( A  e.  V  /\  B  e.  W )  ->  if ( ph ,  A ,  B )  e.  _V )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    /\ wa 370    = wceq 1437    e. wcel 1870   _Vcvv 3087   ifcif 3915
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1665  ax-4 1678  ax-5 1751  ax-6 1797  ax-7 1841  ax-10 1889  ax-11 1894  ax-12 1907  ax-13 2055  ax-ext 2407
This theorem depends on definitions:  df-bi 188  df-or 371  df-an 372  df-tru 1440  df-ex 1660  df-nf 1664  df-sb 1790  df-clab 2415  df-cleq 2421  df-clel 2424  df-nfc 2579  df-rab 2791  df-v 3089  df-un 3447  df-if 3916
This theorem is referenced by:  fsuppmptif  7919  cantnfp1lem1  8182  cantnfp1lem3  8184  symgextfv  17010  pmtrfv  17044  evlslem3  18672  marrepeval  19519  gsummatr01lem3  19613  stdbdmetval  21460  stdbdxmet  21461  ellimc2  22709  psgnfzto1stlem  28452  cdleme31fv  33665  sge0val  37741
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