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Theorem ifexg 3926
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 3861 . . 3  |-  ( x  =  A  ->  if ( ph ,  x ,  y )  =  if ( ph ,  A ,  y ) )
21eleq1d 2451 . 2  |-  ( x  =  A  ->  ( if ( ph ,  x ,  y )  e. 
_V 
<->  if ( ph ,  A ,  y )  e.  _V ) )
3 ifeq2 3862 . . 3  |-  ( y  =  B  ->  if ( ph ,  A , 
y )  =  if ( ph ,  A ,  B ) )
43eleq1d 2451 . 2  |-  ( y  =  B  ->  ( if ( ph ,  A ,  y )  e. 
_V 
<->  if ( ph ,  A ,  B )  e.  _V ) )
5 vex 3037 . . 3  |-  x  e. 
_V
6 vex 3037 . . 3  |-  y  e. 
_V
75, 6ifex 3925 . 2  |-  if (
ph ,  x ,  y )  e.  _V
82, 4, 7vtocl2g 3096 1  |-  ( ( A  e.  V  /\  B  e.  W )  ->  if ( ph ,  A ,  B )  e.  _V )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    /\ wa 367    = wceq 1399    e. wcel 1826   _Vcvv 3034   ifcif 3857
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1626  ax-4 1639  ax-5 1712  ax-6 1755  ax-7 1798  ax-10 1845  ax-11 1850  ax-12 1862  ax-13 2006  ax-ext 2360
This theorem depends on definitions:  df-bi 185  df-or 368  df-an 369  df-tru 1402  df-ex 1621  df-nf 1625  df-sb 1748  df-clab 2368  df-cleq 2374  df-clel 2377  df-nfc 2532  df-rab 2741  df-v 3036  df-un 3394  df-if 3858
This theorem is referenced by:  fsuppmptif  7774  cantnfp1lem1  8010  cantnfp1lem3  8012  cantnfp1lem1OLD  8036  cantnfp1lem3OLD  8038  symgextfv  16560  pmtrfv  16594  evlslem3  18296  marrepeval  19150  gsummatr01lem3  19244  stdbdmetval  21102  stdbdxmet  21103  ellimc2  22366  cdleme31fv  36529
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