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Theorem elif 4505
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 2898 . 2 (if(𝜑, 𝐵, 𝐶) = 𝐵 → (𝐴 ∈ if(𝜑, 𝐵, 𝐶) ↔ 𝐴𝐵))
2 eleq2 2898 . 2 (if(𝜑, 𝐵, 𝐶) = 𝐶 → (𝐴 ∈ if(𝜑, 𝐵, 𝐶) ↔ 𝐴𝐶))
31, 2elimif 4499 1 (𝐴 ∈ if(𝜑, 𝐵, 𝐶) ↔ ((𝜑𝐴𝐵) ∨ (¬ 𝜑𝐴𝐶)))
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wb 207  wa 396  wo 841  wcel 2105  ifcif 4463
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1787  ax-4 1801  ax-5 1902  ax-6 1961  ax-7 2006  ax-8 2107  ax-9 2115  ax-ext 2790
This theorem depends on definitions:  df-bi 208  df-an 397  df-or 842  df-ex 1772  df-sb 2061  df-clab 2797  df-cleq 2811  df-clel 2890  df-if 4464
This theorem is referenced by:  clsk1indlem3  40271
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