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Theorem elimel 3714
 Description: Eliminate a membership hypothesis for weak deduction theorem, when special case B ∈ C is provable. (Contributed by NM, 15-May-1999.)
Hypothesis
Ref Expression
elimel.1 B C
Assertion
Ref Expression
elimel if(A C, A, B) C

Proof of Theorem elimel
StepHypRef Expression
1 eleq1 2413 . 2 (A = if(A C, A, B) → (A C ↔ if(A C, A, B) C))
2 eleq1 2413 . 2 (B = if(A C, A, B) → (B C ↔ if(A C, A, B) C))
3 elimel.1 . 2 B C
41, 2, 3elimhyp 3710 1 if(A C, A, B) C
 Colors of variables: wff setvar class Syntax hints:   ∈ wcel 1710   ifcif 3662 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 3663 This theorem is referenced by: (None)
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