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Theorem ifcl 3698
 Description: Membership (closure) of a conditional operator. (Contributed by NM, 4-Apr-2005.)
Assertion
Ref Expression
ifcl ((A C B C) → if(φ, A, B) C)

Proof of Theorem ifcl
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))
31, 2ifboth 3693 1 ((A C B C) → if(φ, A, B) C)
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ∧ wa 358   ∈ 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:  ifpr  3774
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