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Theorem elimif 4342
Description: Elimination of a conditional operator contained in a wff 𝜓. (Contributed by NM, 15-Feb-2005.) (Proof shortened by NM, 25-Apr-2019.)
Hypotheses
Ref Expression
elimif.1 (if(𝜑, 𝐴, 𝐵) = 𝐴 → (𝜓𝜒))
elimif.2 (if(𝜑, 𝐴, 𝐵) = 𝐵 → (𝜓𝜃))
Assertion
Ref Expression
elimif (𝜓 ↔ ((𝜑𝜒) ∨ (¬ 𝜑𝜃)))

Proof of Theorem elimif
StepHypRef Expression
1 iftrue 4312 . . 3 (𝜑 → if(𝜑, 𝐴, 𝐵) = 𝐴)
2 elimif.1 . . 3 (if(𝜑, 𝐴, 𝐵) = 𝐴 → (𝜓𝜒))
31, 2syl 17 . 2 (𝜑 → (𝜓𝜒))
4 iffalse 4315 . . 3 𝜑 → if(𝜑, 𝐴, 𝐵) = 𝐵)
5 elimif.2 . . 3 (if(𝜑, 𝐴, 𝐵) = 𝐵 → (𝜓𝜃))
64, 5syl 17 . 2 𝜑 → (𝜓𝜃))
73, 6cases 1026 1 (𝜓 ↔ ((𝜑𝜒) ∨ (¬ 𝜑𝜃)))
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wb 198  wa 386  wo 836   = wceq 1601  ifcif 4306
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1839  ax-4 1853  ax-5 1953  ax-6 2021  ax-7 2054  ax-9 2115  ax-10 2134  ax-12 2162  ax-ext 2753
This theorem depends on definitions:  df-bi 199  df-an 387  df-or 837  df-tru 1605  df-ex 1824  df-nf 1828  df-sb 2012  df-clab 2763  df-cleq 2769  df-clel 2773  df-if 4307
This theorem is referenced by:  eqif  4346  elif  4348  ifel  4349  ftc1anclem5  34108  clsk1indlem2  39288
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