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Theorem ifel 4079
 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 2676 . 2 (if(𝜑, 𝐴, 𝐵) = 𝐴 → (if(𝜑, 𝐴, 𝐵) ∈ 𝐶𝐴𝐶))
2 eleq1 2676 . 2 (if(𝜑, 𝐴, 𝐵) = 𝐵 → (if(𝜑, 𝐴, 𝐵) ∈ 𝐶𝐵𝐶))
31, 2elimif 4072 1 (if(𝜑, 𝐴, 𝐵) ∈ 𝐶 ↔ ((𝜑𝐴𝐶) ∨ (¬ 𝜑𝐵𝐶)))
 Colors of variables: wff setvar class Syntax hints:  ¬ wn 3   ↔ wb 195   ∨ wo 382   ∧ wa 383   ∈ wcel 1977  ifcif 4036 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1713  ax-4 1728  ax-5 1827  ax-6 1875  ax-7 1922  ax-10 2006  ax-11 2021  ax-12 2034  ax-13 2234  ax-ext 2590 This theorem depends on definitions:  df-bi 196  df-or 384  df-an 385  df-tru 1478  df-ex 1696  df-nf 1701  df-sb 1868  df-clab 2597  df-cleq 2603  df-clel 2606  df-if 4037 This theorem is referenced by:  clwlkisclwwlklem2a  26313  smatrcl  29190  clsk1independent  37364  clwlkclwwlklem2a  41207
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