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Theorem elimifd 29206
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 431 . . . 4 (𝜓 ∨ ¬ 𝜓)
21biantrur 527 . . 3 (𝜒 ↔ ((𝜓 ∨ ¬ 𝜓) ∧ 𝜒))
32a1i 11 . 2 (𝜑 → (𝜒 ↔ ((𝜓 ∨ ¬ 𝜓) ∧ 𝜒)))
4 andir 911 . . 3 (((𝜓 ∨ ¬ 𝜓) ∧ 𝜒) ↔ ((𝜓𝜒) ∨ (¬ 𝜓𝜒)))
54a1i 11 . 2 (𝜑 → (((𝜓 ∨ ¬ 𝜓) ∧ 𝜒) ↔ ((𝜓𝜒) ∨ (¬ 𝜓𝜒))))
6 iftrue 4064 . . . . 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 670 . . 3 (𝜑 → ((𝜓𝜒) ↔ (𝜓𝜃)))
10 iffalse 4067 . . . . 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 670 . . 3 (𝜑 → ((¬ 𝜓𝜒) ↔ (¬ 𝜓𝜏)))
149, 13orbi12d 745 . 2 (𝜑 → (((𝜓𝜒) ∨ (¬ 𝜓𝜒)) ↔ ((𝜓𝜃) ∨ (¬ 𝜓𝜏))))
153, 5, 143bitrd 294 1 (𝜑 → (𝜒 ↔ ((𝜓𝜃) ∨ (¬ 𝜓𝜏))))
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wb 196  wo 383  wa 384   = wceq 1480  ifcif 4058
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1719  ax-4 1734  ax-5 1836  ax-6 1885  ax-7 1932  ax-9 1996  ax-10 2016  ax-11 2031  ax-12 2044  ax-13 2245  ax-ext 2601
This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-tru 1483  df-ex 1702  df-nf 1707  df-sb 1878  df-clab 2608  df-cleq 2614  df-clel 2617  df-if 4059
This theorem is referenced by:  elim2if  29207
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