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Theorem keepel 3719
 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
StepHypRef 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
51, 2, 3, 4keephyp 3716 1 if(φ, A, B) C
 Colors of variables: wff setvar class Syntax hints:   ∈ wcel 1710   ifcif 3662 This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 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 3663 This theorem is referenced by:  ifex  3720
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