Theorem keepel 3720

Description: Keep a membership hypothesis for weak deduction theorem, when special case B ∈ C is provable. (Contributed by NM, 14-Aug-1999.)

Hypotheses

Ref	Expression
keepel.1	⊢ A ∈ C
keepel.2	⊢ B ∈ C

Assertion

Ref	Expression
keepel	⊢ if(φ, A, B) ∈ C

Proof of Theorem keepel

Step	Hyp	Ref	Expression
1		eleq1 2413	. 2 ⊢ (A = if(φ, A, B) → (A ∈ C ↔ if(φ, A, B) ∈ C))
2		eleq1 2413	. 2 ⊢ (B = if(φ, A, B) → (B ∈ C ↔ if(φ, A, B) ∈ C))
3		keepel.1	. 2 ⊢ A ∈ C
4		keepel.2	. 2 ⊢ B ∈ C
5	1, 2, 3, 4	keephyp 3717	1 ⊢ if(φ, A, B) ∈ C

Syntax hints:   ∈ wcel 1710   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:  ifex  3721