Theorem elim2if 32519

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 → (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))

Assertion

Ref	Expression
elim2if	⊢ (𝜒 ↔ ((𝜑 ∧ 𝜃) ∨ (¬ 𝜑 ∧ ((𝜓 ∧ 𝜏) ∨ (¬ 𝜓 ∧ 𝜂)))))

Proof of Theorem elim2if

Step	Hyp	Ref	Expression
1		iftrue 4481	. . 3 ⊢ (𝜑 → if(𝜑, 𝐴, if(𝜓, 𝐵, 𝐶)) = 𝐴)
2		elim2if.1	. . 3 ⊢ (if(𝜑, 𝐴, if(𝜓, 𝐵, 𝐶)) = 𝐴 → (𝜒 ↔ 𝜃))
3	1, 2	syl 17	. 2 ⊢ (𝜑 → (𝜒 ↔ 𝜃))
4		iffalse 4484	. . . . 5 ⊢ (¬ 𝜑 → if(𝜑, 𝐴, if(𝜓, 𝐵, 𝐶)) = if(𝜓, 𝐵, 𝐶))
5	4	eqeq1d 2733	. . . 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 → (𝜒 ↔ 𝜏))
7	5, 6	biimtrrdi 254	. . 3 ⊢ (¬ 𝜑 → (if(𝜓, 𝐵, 𝐶) = 𝐵 → (𝜒 ↔ 𝜏)))
8	4	eqeq1d 2733	. . . 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 → (𝜒 ↔ 𝜂))
10	8, 9	biimtrrdi 254	. . 3 ⊢ (¬ 𝜑 → (if(𝜓, 𝐵, 𝐶) = 𝐶 → (𝜒 ↔ 𝜂)))
11	7, 10	elimifd 32518	. 2 ⊢ (¬ 𝜑 → (𝜒 ↔ ((𝜓 ∧ 𝜏) ∨ (¬ 𝜓 ∧ 𝜂))))
12	3, 11	cases 1042	1 ⊢ (𝜒 ↔ ((𝜑 ∧ 𝜃) ∨ (¬ 𝜑 ∧ ((𝜓 ∧ 𝜏) ∨ (¬ 𝜓 ∧ 𝜂)))))

Syntax hints:  ¬ wn 3   → wi 4   ↔ wb 206   ∧ wa 395   ∨ wo 847   = wceq 1541  ifcif 4475

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1911  ax-6 1968  ax-7 2009  ax-8 2113  ax-9 2121  ax-ext 2703

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-ex 1781  df-sb 2068  df-clab 2710  df-cleq 2723  df-clel 2806  df-if 4476

This theorem is referenced by:  elim2ifim  32520