Theorem ifexg 3722

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.

Step	Hyp	Ref	Expression
1		ifeq1 3667	. . 3 |- (x = A -> if(ph, x, y) = if(ph, A, y))
2	1	eleq1d 2419	. 2 |- (x = A -> (if(ph, x, y) e. _V <-> if(ph, A, y) e. _V))
3		ifeq2 3668	. . 3 |- (y = B -> if(ph, A, y) = if(ph, A, B))
4	3	eleq1d 2419	. 2 |- (y = B -> (if(ph, A, y) e. _V <-> if(ph, A, B) e. _V))
5		vex 2863	. . 3 |- x e. _V
6		vex 2863	. . 3 |- y e. _V
7	5, 6	ifex 3721	. 2 |- if(ph, x, y) e. _V
8	2, 4, 7	vtocl2g 2919	1 |- ((A e. V /\ B e. W) -> if(ph, A, B) e. _V)

Syntax hints:   -> wi 4   /\ wa 358   = wceq 1642   e. wcel 1710  _Vcvv 2860  ifcif 3663

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1546  ax-5 1557  ax-17 1616  ax-9 1654  ax-8 1675  ax-6 1729  ax-7 1734  ax-11 1746  ax-12 1925  ax-ext 2334

This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-nan 1288  df-tru 1319  df-ex 1542  df-nf 1545  df-sb 1649  df-clab 2340  df-cleq 2346  df-clel 2349  df-nfc 2479  df-rab 2624  df-v 2862  df-nin 3212  df-compl 3213  df-un 3215  df-if 3664

This theorem is referenced by: (None)