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Theorem elim2if 30285
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 4446 . . 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 4449 . . . . 5 𝜑 → if(𝜑, 𝐴, if(𝜓, 𝐵, 𝐶)) = if(𝜓, 𝐵, 𝐶))
54eqeq1d 2823 . . . 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 257 . . 3 𝜑 → (if(𝜓, 𝐵, 𝐶) = 𝐵 → (𝜒𝜏)))
84eqeq1d 2823 . . . 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 257 . . 3 𝜑 → (if(𝜓, 𝐵, 𝐶) = 𝐶 → (𝜒𝜂)))
117, 10elimifd 30284 . 2 𝜑 → (𝜒 ↔ ((𝜓𝜏) ∨ (¬ 𝜓𝜂))))
123, 11cases 1038 1 (𝜒 ↔ ((𝜑𝜃) ∨ (¬ 𝜑 ∧ ((𝜓𝜏) ∨ (¬ 𝜓𝜂)))))
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wb 209  wa 399  wo 844   = wceq 1538  ifcif 4440
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 1912  ax-6 1971  ax-7 2016  ax-8 2117  ax-9 2125  ax-ext 2793
This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-ex 1782  df-sb 2071  df-clab 2800  df-cleq 2814  df-clel 2892  df-if 4441
This theorem is referenced by:  elim2ifim  30286
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