Theorem ifexg 3722

Description: Conditional operator existence. (Contributed by NM, 21-Mar-2011.)

Assertion

Ref	Expression
ifexg	⊢ ((A ∈ V ∧ B ∈ W) → if(φ, A, B) ∈ 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(φ, x, y) = if(φ, A, y))
2	1	eleq1d 2419	. 2 ⊢ (x = A → ( if(φ, x, y) ∈ V ↔ if(φ, A, y) ∈ V))
3		ifeq2 3668	. . 3 ⊢ (y = B → if(φ, A, y) = if(φ, A, B))
4	3	eleq1d 2419	. 2 ⊢ (y = B → ( if(φ, A, y) ∈ V ↔ if(φ, A, B) ∈ V))
5		vex 2863	. . 3 ⊢ x ∈ V
6		vex 2863	. . 3 ⊢ y ∈ V
7	5, 6	ifex 3721	. 2 ⊢ if(φ, x, y) ∈ V
8	2, 4, 7	vtocl2g 2919	1 ⊢ ((A ∈ V ∧ B ∈ W) → if(φ, A, B) ∈ V)

Syntax hints:   → wi 4   ∧ wa 358   = wceq 1642   ∈ 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)