Theorem ifex 1797

Description: Conditional operator existence.

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 1796	1 ⊢ if(φ, A, B) ∈ V

Syntax hints:   ∈ wcel 1092  Vcvv 1348  ifcif 1776

This theorem is referenced by:  oev 3122  unxpdomlem 3649  ruclem13 4897  ruclem15 4899

This theorem was proved from axioms:  ax-1 3  ax-2 4  ax-3 5  ax-mp 6  ax-4 673  ax-5 674  ax-6 675  ax-7 676  ax-gen 677  ax-8 798  ax-9 799  ax-10 800  ax-11 801  ax-12 802  ax-16 922  ax-17 925  ax-ext 1074

This theorem depends on definitions:  df-bi 128  df-or 197  df-an 198  df-ex 679  df-sb 853  df-clab 1093  df-cleq 1097  df-clel 1099  df-if 1777

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