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

Theorem ifexg 3800
 Description: Conditional operator existence. (Contributed by NM, 21-Mar-2011.)
Assertion
Ref Expression
ifexg

Proof of Theorem ifexg
Dummy variables are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 ifeq1 3745 . . 3
21eleq1d 2504 . 2
3 ifeq2 3746 . . 3
43eleq1d 2504 . 2
5 vex 2961 . . 3
6 vex 2961 . . 3
75, 6ifex 3799 . 2
82, 4, 7vtocl2g 3017 1
 Colors of variables: wff set class Syntax hints:   wi 4   wa 360   wceq 1653   wcel 1726  cvv 2958  cif 3741 This theorem is referenced by:  cantnfp1lem1  7636  cantnfp1lem3  7638  stdbdmetval  18546  stdbdxmet  18547  ellimc2  19766  evlslem3  19937  pmtrfv  27374  cdleme31fv  31249 This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1556  ax-5 1567  ax-17 1627  ax-9 1667  ax-8 1688  ax-6 1745  ax-7 1750  ax-11 1762  ax-12 1951  ax-ext 2419 This theorem depends on definitions:  df-bi 179  df-or 361  df-an 362  df-tru 1329  df-ex 1552  df-nf 1555  df-sb 1660  df-clab 2425  df-cleq 2431  df-clel 2434  df-nfc 2563  df-rab 2716  df-v 2960  df-un 3327  df-if 3742
 Copyright terms: Public domain W3C validator