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Theorem elif 3696
 Description: Membership in a conditional operator. (Contributed by NM, 14-Feb-2005.)
Assertion
Ref Expression
elif (A if(φ, B, C) ↔ ((φ A B) φ A C)))

Proof of Theorem elif
StepHypRef Expression
1 eleq2 2414 . 2 ( if(φ, B, C) = B → (A if(φ, B, C) ↔ A B))
2 eleq2 2414 . 2 ( if(φ, B, C) = C → (A if(φ, B, C) ↔ A C))
31, 2elimif 3691 1 (A if(φ, B, C) ↔ ((φ A B) φ A C)))
 Colors of variables: wff setvar class Syntax hints:  ¬ wn 3   ↔ wb 176   ∨ wo 357   ∧ wa 358   ∈ 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:  enprmaplem4  6079
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