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Theorem elim2ifim 29252
 Description: Elimination of two conditional operators for an implication. (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))
elim2ifim.1 (𝜑𝜃)
elim2ifim.2 ((¬ 𝜑𝜓) → 𝜏)
elim2ifim.3 ((¬ 𝜑 ∧ ¬ 𝜓) → 𝜂)
Assertion
Ref Expression
elim2ifim 𝜒

Proof of Theorem elim2ifim
StepHypRef Expression
1 exmid 431 . . 3 (𝜑 ∨ ¬ 𝜑)
2 elim2ifim.1 . . . . 5 (𝜑𝜃)
32ancli 573 . . . 4 (𝜑 → (𝜑𝜃))
4 pm4.42 1003 . . . . . 6 𝜑 ↔ ((¬ 𝜑𝜓) ∨ (¬ 𝜑 ∧ ¬ 𝜓)))
5 elim2ifim.2 . . . . . . . . . 10 ((¬ 𝜑𝜓) → 𝜏)
65ex 450 . . . . . . . . 9 𝜑 → (𝜓𝜏))
76ancld 575 . . . . . . . 8 𝜑 → (𝜓 → (𝜓𝜏)))
87imp 445 . . . . . . 7 ((¬ 𝜑𝜓) → (𝜓𝜏))
9 elim2ifim.3 . . . . . . . . . 10 ((¬ 𝜑 ∧ ¬ 𝜓) → 𝜂)
109ex 450 . . . . . . . . 9 𝜑 → (¬ 𝜓𝜂))
1110ancld 575 . . . . . . . 8 𝜑 → (¬ 𝜓 → (¬ 𝜓𝜂)))
1211imp 445 . . . . . . 7 ((¬ 𝜑 ∧ ¬ 𝜓) → (¬ 𝜓𝜂))
138, 12orim12i 538 . . . . . 6 (((¬ 𝜑𝜓) ∨ (¬ 𝜑 ∧ ¬ 𝜓)) → ((𝜓𝜏) ∨ (¬ 𝜓𝜂)))
144, 13sylbi 207 . . . . 5 𝜑 → ((𝜓𝜏) ∨ (¬ 𝜓𝜂)))
1514ancli 573 . . . 4 𝜑 → (¬ 𝜑 ∧ ((𝜓𝜏) ∨ (¬ 𝜓𝜂))))
163, 15orim12i 538 . . 3 ((𝜑 ∨ ¬ 𝜑) → ((𝜑𝜃) ∨ (¬ 𝜑 ∧ ((𝜓𝜏) ∨ (¬ 𝜓𝜂)))))
171, 16ax-mp 5 . 2 ((𝜑𝜃) ∨ (¬ 𝜑 ∧ ((𝜓𝜏) ∨ (¬ 𝜓𝜂))))
18 elim2if.1 . . 3 (if(𝜑, 𝐴, if(𝜓, 𝐵, 𝐶)) = 𝐴 → (𝜒𝜃))
19 elim2if.2 . . 3 (if(𝜑, 𝐴, if(𝜓, 𝐵, 𝐶)) = 𝐵 → (𝜒𝜏))
20 elim2if.3 . . 3 (if(𝜑, 𝐴, if(𝜓, 𝐵, 𝐶)) = 𝐶 → (𝜒𝜂))
2118, 19, 20elim2if 29251 . 2 (𝜒 ↔ ((𝜑𝜃) ∨ (¬ 𝜑 ∧ ((𝜓𝜏) ∨ (¬ 𝜓𝜂)))))
2217, 21mpbir 221 1 𝜒
 Colors of variables: wff setvar class Syntax hints:  ¬ wn 3   → wi 4   ↔ wb 196   ∨ wo 383   ∧ wa 384   = wceq 1480  ifcif 4064 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 4065 This theorem is referenced by: (None)
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