Theorem ifex 3721

Description: Conditional operator existence. (Contributed by NM, 2-Sep-2004.)

Hypotheses

Ref	Expression
dedex.1	⊢ A ∈ V
dedex.2	⊢ B ∈ V

Assertion

Ref	Expression
ifex	⊢ if(φ, A, B) ∈ V

Proof of Theorem ifex

Step	Hyp	Ref	Expression
1		dedex.1	. 2 ⊢ A ∈ V
2		dedex.2	. 2 ⊢ B ∈ V
3	1, 2	keepel 3720	1 ⊢ if(φ, A, B) ∈ V

Syntax hints:   ∈ 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-tru 1319  df-ex 1542  df-nf 1545  df-sb 1649  df-clab 2340  df-cleq 2346  df-clel 2349  df-if 3664

This theorem is referenced by:  ifexg  3722  setswithex  4323  tfinex  4486  enprmaplem5  6081