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Theorem elif 4569
Description: Membership in a conditional operator. (Contributed by NM, 14-Feb-2005.)
Assertion
Ref Expression
elif (𝐴 ∈ if(𝜑, 𝐵, 𝐶) ↔ ((𝜑𝐴𝐵) ∨ (¬ 𝜑𝐴𝐶)))

Proof of Theorem elif
StepHypRef Expression
1 eleq2 2830 . 2 (if(𝜑, 𝐵, 𝐶) = 𝐵 → (𝐴 ∈ if(𝜑, 𝐵, 𝐶) ↔ 𝐴𝐵))
2 eleq2 2830 . 2 (if(𝜑, 𝐵, 𝐶) = 𝐶 → (𝐴 ∈ if(𝜑, 𝐵, 𝐶) ↔ 𝐴𝐶))
31, 2elimif 4563 1 (𝐴 ∈ if(𝜑, 𝐵, 𝐶) ↔ ((𝜑𝐴𝐵) ∨ (¬ 𝜑𝐴𝐶)))
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wb 206  wa 395  wo 848  wcel 2108  ifcif 4525
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-ext 2708
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-ex 1780  df-sb 2065  df-clab 2715  df-cleq 2729  df-clel 2816  df-if 4526
This theorem is referenced by:  bj-imdirco  37191  safesnsupfiss  43428  clsk1indlem3  44056
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