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Theorem ifel 4565
Description: Membership of a conditional operator. (Contributed by NM, 10-Sep-2005.)
Assertion
Ref Expression
ifel (if(𝜑, 𝐴, 𝐵) ∈ 𝐶 ↔ ((𝜑𝐴𝐶) ∨ (¬ 𝜑𝐵𝐶)))

Proof of Theorem ifel
StepHypRef Expression
1 eleq1 2813 . 2 (if(𝜑, 𝐴, 𝐵) = 𝐴 → (if(𝜑, 𝐴, 𝐵) ∈ 𝐶𝐴𝐶))
2 eleq1 2813 . 2 (if(𝜑, 𝐴, 𝐵) = 𝐵 → (if(𝜑, 𝐴, 𝐵) ∈ 𝐶𝐵𝐶))
31, 2elimif 4558 1 (if(𝜑, 𝐴, 𝐵) ∈ 𝐶 ↔ ((𝜑𝐴𝐶) ∨ (¬ 𝜑𝐵𝐶)))
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wb 205  wa 395  wo 844  wcel 2098  ifcif 4521
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1905  ax-6 1963  ax-7 2003  ax-8 2100  ax-9 2108  ax-ext 2695
This theorem depends on definitions:  df-bi 206  df-an 396  df-or 845  df-ex 1774  df-sb 2060  df-clab 2702  df-cleq 2716  df-clel 2802  df-if 4522
This theorem is referenced by:  clwlkclwwlklem2a  29723  smatrcl  33268  clsk1independent  43311
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