MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  ifexg Unicode version

Theorem ifexg 3637
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 3582 . . 3  |-  ( x  =  A  ->  if ( ph ,  x ,  y )  =  if ( ph ,  A ,  y ) )
21eleq1d 2362 . 2  |-  ( x  =  A  ->  ( if ( ph ,  x ,  y )  e. 
_V 
<->  if ( ph ,  A ,  y )  e.  _V ) )
3 ifeq2 3583 . . 3  |-  ( y  =  B  ->  if ( ph ,  A , 
y )  =  if ( ph ,  A ,  B ) )
43eleq1d 2362 . 2  |-  ( y  =  B  ->  ( if ( ph ,  A ,  y )  e. 
_V 
<->  if ( ph ,  A ,  B )  e.  _V ) )
5 vex 2804 . . 3  |-  x  e. 
_V
6 vex 2804 . . 3  |-  y  e. 
_V
75, 6ifex 3636 . 2  |-  if (
ph ,  x ,  y )  e.  _V
82, 4, 7vtocl2g 2860 1  |-  ( ( A  e.  V  /\  B  e.  W )  ->  if ( ph ,  A ,  B )  e.  _V )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 358    = wceq 1632    e. wcel 1696   _Vcvv 2801   ifcif 3578
This theorem is referenced by:  cantnfp1lem1  7396  cantnfp1lem3  7398  stdbdmetval  18076  stdbdxmet  18077  ellimc2  19243  evlslem3  19414  pmtrfv  27498  cdleme31fv  31201
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1536  ax-5 1547  ax-17 1606  ax-9 1644  ax-8 1661  ax-6 1715  ax-7 1720  ax-11 1727  ax-12 1878  ax-ext 2277
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-tru 1310  df-ex 1532  df-nf 1535  df-sb 1639  df-clab 2283  df-cleq 2289  df-clel 2292  df-nfc 2421  df-rab 2565  df-v 2803  df-un 3170  df-if 3579
  Copyright terms: Public domain W3C validator