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Theorem elim2if 29347
Description: Elimination of two conditional operators contained in a wff 𝜒. (Contributed by Thierry Arnoux, 25-Jan-2017.)
Hypotheses
Ref Expression
elim2if.1 (if(𝜑, 𝐴, if(𝜓, 𝐵, 𝐶)) = 𝐴 → (𝜒𝜃))
elim2if.2 (if(𝜑, 𝐴, if(𝜓, 𝐵, 𝐶)) = 𝐵 → (𝜒𝜏))
elim2if.3 (if(𝜑, 𝐴, if(𝜓, 𝐵, 𝐶)) = 𝐶 → (𝜒𝜂))
Assertion
Ref Expression
elim2if (𝜒 ↔ ((𝜑𝜃) ∨ (¬ 𝜑 ∧ ((𝜓𝜏) ∨ (¬ 𝜓𝜂)))))

Proof of Theorem elim2if
StepHypRef Expression
1 iftrue 4090 . . 3 (𝜑 → if(𝜑, 𝐴, if(𝜓, 𝐵, 𝐶)) = 𝐴)
2 elim2if.1 . . 3 (if(𝜑, 𝐴, if(𝜓, 𝐵, 𝐶)) = 𝐴 → (𝜒𝜃))
31, 2syl 17 . 2 (𝜑 → (𝜒𝜃))
4 iffalse 4093 . . . . 5 𝜑 → if(𝜑, 𝐴, if(𝜓, 𝐵, 𝐶)) = if(𝜓, 𝐵, 𝐶))
54eqeq1d 2623 . . . 4 𝜑 → (if(𝜑, 𝐴, if(𝜓, 𝐵, 𝐶)) = 𝐵 ↔ if(𝜓, 𝐵, 𝐶) = 𝐵))
6 elim2if.2 . . . 4 (if(𝜑, 𝐴, if(𝜓, 𝐵, 𝐶)) = 𝐵 → (𝜒𝜏))
75, 6syl6bir 244 . . 3 𝜑 → (if(𝜓, 𝐵, 𝐶) = 𝐵 → (𝜒𝜏)))
84eqeq1d 2623 . . . 4 𝜑 → (if(𝜑, 𝐴, if(𝜓, 𝐵, 𝐶)) = 𝐶 ↔ if(𝜓, 𝐵, 𝐶) = 𝐶))
9 elim2if.3 . . . 4 (if(𝜑, 𝐴, if(𝜓, 𝐵, 𝐶)) = 𝐶 → (𝜒𝜂))
108, 9syl6bir 244 . . 3 𝜑 → (if(𝜓, 𝐵, 𝐶) = 𝐶 → (𝜒𝜂)))
117, 10elimifd 29346 . 2 𝜑 → (𝜒 ↔ ((𝜓𝜏) ∨ (¬ 𝜓𝜂))))
123, 11cases 992 1 (𝜒 ↔ ((𝜑𝜃) ∨ (¬ 𝜑 ∧ ((𝜓𝜏) ∨ (¬ 𝜓𝜂)))))
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wb 196  wo 383  wa 384   = wceq 1482  ifcif 4084
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1721  ax-4 1736  ax-5 1838  ax-6 1887  ax-7 1934  ax-9 1998  ax-10 2018  ax-11 2033  ax-12 2046  ax-13 2245  ax-ext 2601
This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-tru 1485  df-ex 1704  df-nf 1709  df-sb 1880  df-clab 2608  df-cleq 2614  df-clel 2617  df-if 4085
This theorem is referenced by:  elim2ifim  29348
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