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Theorem elimifd 30313
Description: Elimination of a conditional operator contained in a wff 𝜒. (Contributed by Thierry Arnoux, 25-Jan-2017.)
Hypotheses
Ref Expression
elimifd.1 (𝜑 → (if(𝜓, 𝐴, 𝐵) = 𝐴 → (𝜒𝜃)))
elimifd.2 (𝜑 → (if(𝜓, 𝐴, 𝐵) = 𝐵 → (𝜒𝜏)))
Assertion
Ref Expression
elimifd (𝜑 → (𝜒 ↔ ((𝜓𝜃) ∨ (¬ 𝜓𝜏))))

Proof of Theorem elimifd
StepHypRef Expression
1 exmid 892 . . . 4 (𝜓 ∨ ¬ 𝜓)
21biantrur 534 . . 3 (𝜒 ↔ ((𝜓 ∨ ¬ 𝜓) ∧ 𝜒))
32a1i 11 . 2 (𝜑 → (𝜒 ↔ ((𝜓 ∨ ¬ 𝜓) ∧ 𝜒)))
4 andir 1006 . . 3 (((𝜓 ∨ ¬ 𝜓) ∧ 𝜒) ↔ ((𝜓𝜒) ∨ (¬ 𝜓𝜒)))
54a1i 11 . 2 (𝜑 → (((𝜓 ∨ ¬ 𝜓) ∧ 𝜒) ↔ ((𝜓𝜒) ∨ (¬ 𝜓𝜒))))
6 iftrue 4434 . . . . 5 (𝜓 → if(𝜓, 𝐴, 𝐵) = 𝐴)
7 elimifd.1 . . . . 5 (𝜑 → (if(𝜓, 𝐴, 𝐵) = 𝐴 → (𝜒𝜃)))
86, 7syl5 34 . . . 4 (𝜑 → (𝜓 → (𝜒𝜃)))
98pm5.32d 580 . . 3 (𝜑 → ((𝜓𝜒) ↔ (𝜓𝜃)))
10 iffalse 4437 . . . . 5 𝜓 → if(𝜓, 𝐴, 𝐵) = 𝐵)
11 elimifd.2 . . . . 5 (𝜑 → (if(𝜓, 𝐴, 𝐵) = 𝐵 → (𝜒𝜏)))
1210, 11syl5 34 . . . 4 (𝜑 → (¬ 𝜓 → (𝜒𝜏)))
1312pm5.32d 580 . . 3 (𝜑 → ((¬ 𝜓𝜒) ↔ (¬ 𝜓𝜏)))
149, 13orbi12d 916 . 2 (𝜑 → (((𝜓𝜒) ∨ (¬ 𝜓𝜒)) ↔ ((𝜓𝜃) ∨ (¬ 𝜓𝜏))))
153, 5, 143bitrd 308 1 (𝜑 → (𝜒 ↔ ((𝜓𝜃) ∨ (¬ 𝜓𝜏))))
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wb 209  wa 399  wo 844   = wceq 1538  ifcif 4428
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1911  ax-6 1970  ax-7 2015  ax-8 2114  ax-9 2122  ax-ext 2773
This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-ex 1782  df-sb 2070  df-clab 2780  df-cleq 2794  df-clel 2873  df-if 4429
This theorem is referenced by:  elim2if  30314
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