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Theorem elim2ifim 29696
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 860 . . 3 (𝜑 ∨ ¬ 𝜑)
2 elim2ifim.1 . . . . 5 (𝜑𝜃)
32ancli 530 . . . 4 (𝜑 → (𝜑𝜃))
4 pm4.42 1040 . . . . . 6 𝜑 ↔ ((¬ 𝜑𝜓) ∨ (¬ 𝜑 ∧ ¬ 𝜓)))
5 elim2ifim.2 . . . . . . . . . 10 ((¬ 𝜑𝜓) → 𝜏)
65ex 397 . . . . . . . . 9 𝜑 → (𝜓𝜏))
76ancld 532 . . . . . . . 8 𝜑 → (𝜓 → (𝜓𝜏)))
87imp 393 . . . . . . 7 ((¬ 𝜑𝜓) → (𝜓𝜏))
9 elim2ifim.3 . . . . . . . . . 10 ((¬ 𝜑 ∧ ¬ 𝜓) → 𝜂)
109ex 397 . . . . . . . . 9 𝜑 → (¬ 𝜓𝜂))
1110ancld 532 . . . . . . . 8 𝜑 → (¬ 𝜓 → (¬ 𝜓𝜂)))
1211imp 393 . . . . . . 7 ((¬ 𝜑 ∧ ¬ 𝜓) → (¬ 𝜓𝜂))
138, 12orim12i 874 . . . . . 6 (((¬ 𝜑𝜓) ∨ (¬ 𝜑 ∧ ¬ 𝜓)) → ((𝜓𝜏) ∨ (¬ 𝜓𝜂)))
144, 13sylbi 207 . . . . 5 𝜑 → ((𝜓𝜏) ∨ (¬ 𝜓𝜂)))
1514ancli 530 . . . 4 𝜑 → (¬ 𝜑 ∧ ((𝜓𝜏) ∨ (¬ 𝜓𝜂))))
163, 15orim12i 874 . . 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 29695 . 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  wa 382  wo 826   = wceq 1630  ifcif 4223
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1869  ax-4 1884  ax-5 1990  ax-6 2056  ax-7 2092  ax-9 2153  ax-10 2173  ax-11 2189  ax-12 2202  ax-ext 2750
This theorem depends on definitions:  df-bi 197  df-an 383  df-or 827  df-tru 1633  df-ex 1852  df-nf 1857  df-sb 2049  df-clab 2757  df-cleq 2763  df-clel 2766  df-if 4224
This theorem is referenced by: (None)
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