Theorem 2if2dc 3677

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)
2if2dc.ps	⊢ (𝜑 → DECID 𝜓)
2if2dc.th	⊢ ((𝜑 ∧ ¬ 𝜓) → DECID 𝜃)

Assertion

Ref	Expression
2if2dc	⊢ (𝜑 → 𝐷 = if(𝜓, 𝐴, if(𝜃, 𝐵, 𝐶)))

Proof of Theorem 2if2dc

Step	Hyp	Ref	Expression
1		2if2.1	. . 3 ⊢ ((𝜑 ∧ 𝜓) → 𝐷 = 𝐴)
2		iftrue 3642	. . . 4 ⊢ (𝜓 → if(𝜓, 𝐴, if(𝜃, 𝐵, 𝐶)) = 𝐴)
3	2	adantl 277	. . 3 ⊢ ((𝜑 ∧ 𝜓) → if(𝜓, 𝐴, if(𝜃, 𝐵, 𝐶)) = 𝐴)
4	1, 3	eqtr4d 2274	. 2 ⊢ ((𝜑 ∧ 𝜓) → 𝐷 = if(𝜓, 𝐴, if(𝜃, 𝐵, 𝐶)))
5		2if2.2	. . . . . 6 ⊢ ((𝜑 ∧ ¬ 𝜓 ∧ 𝜃) → 𝐷 = 𝐵)
6	5	3expa 1234	. . . . 5 ⊢ (((𝜑 ∧ ¬ 𝜓) ∧ 𝜃) → 𝐷 = 𝐵)
7		iftrue 3642	. . . . . 6 ⊢ (𝜃 → if(𝜃, 𝐵, 𝐶) = 𝐵)
8	7	adantl 277	. . . . 5 ⊢ (((𝜑 ∧ ¬ 𝜓) ∧ 𝜃) → if(𝜃, 𝐵, 𝐶) = 𝐵)
9	6, 8	eqtr4d 2274	. . . 4 ⊢ (((𝜑 ∧ ¬ 𝜓) ∧ 𝜃) → 𝐷 = if(𝜃, 𝐵, 𝐶))
10		2if2.3	. . . . . 6 ⊢ ((𝜑 ∧ ¬ 𝜓 ∧ ¬ 𝜃) → 𝐷 = 𝐶)
11	10	3expa 1234	. . . . 5 ⊢ (((𝜑 ∧ ¬ 𝜓) ∧ ¬ 𝜃) → 𝐷 = 𝐶)
12		iffalse 3645	. . . . . . 7 ⊢ (¬ 𝜃 → if(𝜃, 𝐵, 𝐶) = 𝐶)
13	12	eqcomd 2244	. . . . . 6 ⊢ (¬ 𝜃 → 𝐶 = if(𝜃, 𝐵, 𝐶))
14	13	adantl 277	. . . . 5 ⊢ (((𝜑 ∧ ¬ 𝜓) ∧ ¬ 𝜃) → 𝐶 = if(𝜃, 𝐵, 𝐶))
15	11, 14	eqtrd 2271	. . . 4 ⊢ (((𝜑 ∧ ¬ 𝜓) ∧ ¬ 𝜃) → 𝐷 = if(𝜃, 𝐵, 𝐶))
16		2if2dc.th	. . . . 5 ⊢ ((𝜑 ∧ ¬ 𝜓) → DECID 𝜃)
17		exmiddc 848	. . . . 5 ⊢ (DECID 𝜃 → (𝜃 ∨ ¬ 𝜃))
18	16, 17	syl 14	. . . 4 ⊢ ((𝜑 ∧ ¬ 𝜓) → (𝜃 ∨ ¬ 𝜃))
19	9, 15, 18	mpjaodan 810	. . 3 ⊢ ((𝜑 ∧ ¬ 𝜓) → 𝐷 = if(𝜃, 𝐵, 𝐶))
20		iffalse 3645	. . . 4 ⊢ (¬ 𝜓 → if(𝜓, 𝐴, if(𝜃, 𝐵, 𝐶)) = if(𝜃, 𝐵, 𝐶))
21	20	adantl 277	. . 3 ⊢ ((𝜑 ∧ ¬ 𝜓) → if(𝜓, 𝐴, if(𝜃, 𝐵, 𝐶)) = if(𝜃, 𝐵, 𝐶))
22	19, 21	eqtr4d 2274	. 2 ⊢ ((𝜑 ∧ ¬ 𝜓) → 𝐷 = if(𝜓, 𝐴, if(𝜃, 𝐵, 𝐶)))
23		2if2dc.ps	. . 3 ⊢ (𝜑 → DECID 𝜓)
24		exmiddc 848	. . 3 ⊢ (DECID 𝜓 → (𝜓 ∨ ¬ 𝜓))
25	23, 24	syl 14	. 2 ⊢ (𝜑 → (𝜓 ∨ ¬ 𝜓))
26	4, 22, 25	mpjaodan 810	1 ⊢ (𝜑 → 𝐷 = if(𝜓, 𝐴, if(𝜃, 𝐵, 𝐶)))

Syntax hints:  ¬ wn 3   → wi 4   ∧ wa 104   ∨ wo 720  DECID wdc 846   ∧ w3a 1009   = wceq 1402  ifcif 3635

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-in2 624  ax-io 721  ax-5 1500  ax-7 1501  ax-gen 1502  ax-ie1 1546  ax-ie2 1547  ax-8 1557  ax-11 1559  ax-4 1563  ax-17 1579  ax-i9 1583  ax-ial 1587  ax-i5r 1588  ax-ext 2220

This theorem depends on definitions:  df-bi 117  df-dc 847  df-3an 1011  df-nf 1514  df-sb 1816  df-clab 2225  df-cleq 2231  df-clel 2234  df-if 3636

This theorem is referenced by:  pfxccat3  11484  swrdccat  11485  swrdccat3b  11490