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Theorem elimif 4072
 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 4042 . . 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 4045 . . 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 1004 1 (𝜓 ↔ ((𝜑𝜒) ∨ (¬ 𝜑𝜃)))
 Colors of variables: wff setvar class Syntax hints:  ¬ wn 3   → wi 4   ↔ wb 195   ∨ wo 382   ∧ wa 383   = wceq 1475  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:  eqif  4076  elif  4078  ifel  4079  ftc1anclem5  32659  clsk1indlem2  37360
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