Theorem elim2ifim 30789

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 → (𝜒 ↔ 𝜃))
elim2if.2	⊢ (if(𝜑, 𝐴, if(𝜓, 𝐵, 𝐶)) = 𝐵 → (𝜒 ↔ 𝜏))
elim2if.3	⊢ (if(𝜑, 𝐴, if(𝜓, 𝐵, 𝐶)) = 𝐶 → (𝜒 ↔ 𝜂))
elim2ifim.1	⊢ (𝜑 → 𝜃)
elim2ifim.2	⊢ ((¬ 𝜑 ∧ 𝜓) → 𝜏)
elim2ifim.3	⊢ ((¬ 𝜑 ∧ ¬ 𝜓) → 𝜂)

Assertion

Ref	Expression
elim2ifim	⊢ 𝜒

Proof of Theorem elim2ifim

Step	Hyp	Ref	Expression
1		exmid 891	. . 3 ⊢ (𝜑 ∨ ¬ 𝜑)
2		elim2ifim.1	. . . . 5 ⊢ (𝜑 → 𝜃)
3	2	ancli 548	. . . 4 ⊢ (𝜑 → (𝜑 ∧ 𝜃))
4		pm4.42 1050	. . . . . 6 ⊢ (¬ 𝜑 ↔ ((¬ 𝜑 ∧ 𝜓) ∨ (¬ 𝜑 ∧ ¬ 𝜓)))
5		elim2ifim.2	. . . . . . . . . 10 ⊢ ((¬ 𝜑 ∧ 𝜓) → 𝜏)
6	5	ex 412	. . . . . . . . 9 ⊢ (¬ 𝜑 → (𝜓 → 𝜏))
7	6	ancld 550	. . . . . . . 8 ⊢ (¬ 𝜑 → (𝜓 → (𝜓 ∧ 𝜏)))
8	7	imp 406	. . . . . . 7 ⊢ ((¬ 𝜑 ∧ 𝜓) → (𝜓 ∧ 𝜏))
9		elim2ifim.3	. . . . . . . . . 10 ⊢ ((¬ 𝜑 ∧ ¬ 𝜓) → 𝜂)
10	9	ex 412	. . . . . . . . 9 ⊢ (¬ 𝜑 → (¬ 𝜓 → 𝜂))
11	10	ancld 550	. . . . . . . 8 ⊢ (¬ 𝜑 → (¬ 𝜓 → (¬ 𝜓 ∧ 𝜂)))
12	11	imp 406	. . . . . . 7 ⊢ ((¬ 𝜑 ∧ ¬ 𝜓) → (¬ 𝜓 ∧ 𝜂))
13	8, 12	orim12i 905	. . . . . 6 ⊢ (((¬ 𝜑 ∧ 𝜓) ∨ (¬ 𝜑 ∧ ¬ 𝜓)) → ((𝜓 ∧ 𝜏) ∨ (¬ 𝜓 ∧ 𝜂)))
14	4, 13	sylbi 216	. . . . 5 ⊢ (¬ 𝜑 → ((𝜓 ∧ 𝜏) ∨ (¬ 𝜓 ∧ 𝜂)))
15	14	ancli 548	. . . 4 ⊢ (¬ 𝜑 → (¬ 𝜑 ∧ ((𝜓 ∧ 𝜏) ∨ (¬ 𝜓 ∧ 𝜂))))
16	3, 15	orim12i 905	. . 3 ⊢ ((𝜑 ∨ ¬ 𝜑) → ((𝜑 ∧ 𝜃) ∨ (¬ 𝜑 ∧ ((𝜓 ∧ 𝜏) ∨ (¬ 𝜓 ∧ 𝜂)))))
17	1, 16	ax-mp 5	. 2 ⊢ ((𝜑 ∧ 𝜃) ∨ (¬ 𝜑 ∧ ((𝜓 ∧ 𝜏) ∨ (¬ 𝜓 ∧ 𝜂))))
18		elim2if.1	. . 3 ⊢ (if(𝜑, 𝐴, if(𝜓, 𝐵, 𝐶)) = 𝐴 → (𝜒 ↔ 𝜃))
19		elim2if.2	. . 3 ⊢ (if(𝜑, 𝐴, if(𝜓, 𝐵, 𝐶)) = 𝐵 → (𝜒 ↔ 𝜏))
20		elim2if.3	. . 3 ⊢ (if(𝜑, 𝐴, if(𝜓, 𝐵, 𝐶)) = 𝐶 → (𝜒 ↔ 𝜂))
21	18, 19, 20	elim2if 30788	. 2 ⊢ (𝜒 ↔ ((𝜑 ∧ 𝜃) ∨ (¬ 𝜑 ∧ ((𝜓 ∧ 𝜏) ∨ (¬ 𝜓 ∧ 𝜂)))))
22	17, 21	mpbir 230	1 ⊢ 𝜒

Syntax hints:  ¬ wn 3   → wi 4   ↔ wb 205   ∧ wa 395   ∨ wo 843   = wceq 1539  ifcif 4456

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1799  ax-4 1813  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2110  ax-9 2118  ax-ext 2709

This theorem depends on definitions:  df-bi 206  df-an 396  df-or 844  df-ex 1784  df-sb 2069  df-clab 2716  df-cleq 2730  df-clel 2817  df-if 4457

This theorem is referenced by: (None)