Theorem 2if2 4511

Description: Resolve two nested conditionals. (Contributed by Alexander van der Vekens, 27-Mar-2018.)

Hypotheses

Ref	Expression
2if2.1	⊢ ((𝜑 ∧ 𝜓) → 𝐷 = 𝐴)
2if2.2	⊢ ((𝜑 ∧ ¬ 𝜓 ∧ 𝜃) → 𝐷 = 𝐵)
2if2.3	⊢ ((𝜑 ∧ ¬ 𝜓 ∧ ¬ 𝜃) → 𝐷 = 𝐶)

Assertion

Ref	Expression
2if2	⊢ (𝜑 → 𝐷 = if(𝜓, 𝐴, if(𝜃, 𝐵, 𝐶)))

Proof of Theorem 2if2

Step	Hyp	Ref	Expression
1		2if2.1	. . 3 ⊢ ((𝜑 ∧ 𝜓) → 𝐷 = 𝐴)
2		iftrue 4462	. . . 4 ⊢ (𝜓 → if(𝜓, 𝐴, if(𝜃, 𝐵, 𝐶)) = 𝐴)
3	2	adantl 481	. . 3 ⊢ ((𝜑 ∧ 𝜓) → if(𝜓, 𝐴, if(𝜃, 𝐵, 𝐶)) = 𝐴)
4	1, 3	eqtr4d 2781	. 2 ⊢ ((𝜑 ∧ 𝜓) → 𝐷 = if(𝜓, 𝐴, if(𝜃, 𝐵, 𝐶)))
5		2if2.2	. . . . . 6 ⊢ ((𝜑 ∧ ¬ 𝜓 ∧ 𝜃) → 𝐷 = 𝐵)
6	5	3expa 1116	. . . . 5 ⊢ (((𝜑 ∧ ¬ 𝜓) ∧ 𝜃) → 𝐷 = 𝐵)
7		iftrue 4462	. . . . . 6 ⊢ (𝜃 → if(𝜃, 𝐵, 𝐶) = 𝐵)
8	7	adantl 481	. . . . 5 ⊢ (((𝜑 ∧ ¬ 𝜓) ∧ 𝜃) → if(𝜃, 𝐵, 𝐶) = 𝐵)
9	6, 8	eqtr4d 2781	. . . 4 ⊢ (((𝜑 ∧ ¬ 𝜓) ∧ 𝜃) → 𝐷 = if(𝜃, 𝐵, 𝐶))
10		2if2.3	. . . . . 6 ⊢ ((𝜑 ∧ ¬ 𝜓 ∧ ¬ 𝜃) → 𝐷 = 𝐶)
11	10	3expa 1116	. . . . 5 ⊢ (((𝜑 ∧ ¬ 𝜓) ∧ ¬ 𝜃) → 𝐷 = 𝐶)
12		iffalse 4465	. . . . . . 7 ⊢ (¬ 𝜃 → if(𝜃, 𝐵, 𝐶) = 𝐶)
13	12	eqcomd 2744	. . . . . 6 ⊢ (¬ 𝜃 → 𝐶 = if(𝜃, 𝐵, 𝐶))
14	13	adantl 481	. . . . 5 ⊢ (((𝜑 ∧ ¬ 𝜓) ∧ ¬ 𝜃) → 𝐶 = if(𝜃, 𝐵, 𝐶))
15	11, 14	eqtrd 2778	. . . 4 ⊢ (((𝜑 ∧ ¬ 𝜓) ∧ ¬ 𝜃) → 𝐷 = if(𝜃, 𝐵, 𝐶))
16	9, 15	pm2.61dan 809	. . 3 ⊢ ((𝜑 ∧ ¬ 𝜓) → 𝐷 = if(𝜃, 𝐵, 𝐶))
17		iffalse 4465	. . . 4 ⊢ (¬ 𝜓 → if(𝜓, 𝐴, if(𝜃, 𝐵, 𝐶)) = if(𝜃, 𝐵, 𝐶))
18	17	adantl 481	. . 3 ⊢ ((𝜑 ∧ ¬ 𝜓) → if(𝜓, 𝐴, if(𝜃, 𝐵, 𝐶)) = if(𝜃, 𝐵, 𝐶))
19	16, 18	eqtr4d 2781	. 2 ⊢ ((𝜑 ∧ ¬ 𝜓) → 𝐷 = if(𝜓, 𝐴, if(𝜃, 𝐵, 𝐶)))
20	4, 19	pm2.61dan 809	1 ⊢ (𝜑 → 𝐷 = if(𝜓, 𝐴, if(𝜃, 𝐵, 𝐶)))

Syntax hints:  ¬ wn 3   → wi 4   ∧ wa 395   ∧ w3a 1085   = 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-3an 1087  df-ex 1784  df-sb 2069  df-clab 2716  df-cleq 2730  df-clel 2817  df-if 4457

This theorem is referenced by:  pfxccat3  14375  swrdccat  14376  swrdccat3b  14381