Theorem or3or 41520

Description: Decompose disjunction into three cases. (Contributed by RP, 5-Jul-2021.)

Assertion

Ref	Expression
or3or	⊢ ((𝜑 ∨ 𝜓) ↔ ((𝜑 ∧ 𝜓) ∨ (𝜑 ∧ ¬ 𝜓) ∨ (¬ 𝜑 ∧ 𝜓)))

Proof of Theorem or3or

Step	Hyp	Ref	Expression
1		excxor 1509	. . 3 ⊢ ((𝜑 ⊻ 𝜓) ↔ ((𝜑 ∧ ¬ 𝜓) ∨ (¬ 𝜑 ∧ 𝜓)))
2	1	orbi2i 909	. 2 ⊢ (((𝜑 ∧ 𝜓) ∨ (𝜑 ⊻ 𝜓)) ↔ ((𝜑 ∧ 𝜓) ∨ ((𝜑 ∧ ¬ 𝜓) ∨ (¬ 𝜑 ∧ 𝜓))))
3		orc 863	. . . 4 ⊢ (𝜑 → (𝜑 ∨ 𝜓))
4		exmid 891	. . . . 5 ⊢ (𝜓 ∨ ¬ 𝜓)
5		pm3.2 469	. . . . . 6 ⊢ (𝜑 → (𝜓 → (𝜑 ∧ 𝜓)))
6		biimp 214	. . . . . . . . . 10 ⊢ ((𝜑 ↔ 𝜓) → (𝜑 → 𝜓))
7		iman 401	. . . . . . . . . 10 ⊢ ((𝜑 → 𝜓) ↔ ¬ (𝜑 ∧ ¬ 𝜓))
8	6, 7	sylib 217	. . . . . . . . 9 ⊢ ((𝜑 ↔ 𝜓) → ¬ (𝜑 ∧ ¬ 𝜓))
9	8	con2i 139	. . . . . . . 8 ⊢ ((𝜑 ∧ ¬ 𝜓) → ¬ (𝜑 ↔ 𝜓))
10	9	ex 412	. . . . . . 7 ⊢ (𝜑 → (¬ 𝜓 → ¬ (𝜑 ↔ 𝜓)))
11		df-xor 1504	. . . . . . . 8 ⊢ ((𝜑 ⊻ 𝜓) ↔ ¬ (𝜑 ↔ 𝜓))
12	11	bicomi 223	. . . . . . 7 ⊢ (¬ (𝜑 ↔ 𝜓) ↔ (𝜑 ⊻ 𝜓))
13	10, 12	syl6ib 250	. . . . . 6 ⊢ (𝜑 → (¬ 𝜓 → (𝜑 ⊻ 𝜓)))
14	5, 13	orim12d 961	. . . . 5 ⊢ (𝜑 → ((𝜓 ∨ ¬ 𝜓) → ((𝜑 ∧ 𝜓) ∨ (𝜑 ⊻ 𝜓))))
15	4, 14	mpi 20	. . . 4 ⊢ (𝜑 → ((𝜑 ∧ 𝜓) ∨ (𝜑 ⊻ 𝜓)))
16	3, 15	2thd 264	. . 3 ⊢ (𝜑 → ((𝜑 ∨ 𝜓) ↔ ((𝜑 ∧ 𝜓) ∨ (𝜑 ⊻ 𝜓))))
17		bicom 221	. . . . . . 7 ⊢ ((𝜑 ↔ 𝜓) ↔ (𝜓 ↔ 𝜑))
18		bibif 371	. . . . . . 7 ⊢ (¬ 𝜑 → ((𝜓 ↔ 𝜑) ↔ ¬ 𝜓))
19	17, 18	syl5bb 282	. . . . . 6 ⊢ (¬ 𝜑 → ((𝜑 ↔ 𝜓) ↔ ¬ 𝜓))
20	19	con2bid 354	. . . . 5 ⊢ (¬ 𝜑 → (𝜓 ↔ ¬ (𝜑 ↔ 𝜓)))
21	20, 12	bitrdi 286	. . . 4 ⊢ (¬ 𝜑 → (𝜓 ↔ (𝜑 ⊻ 𝜓)))
22		biorf 933	. . . 4 ⊢ (¬ 𝜑 → (𝜓 ↔ (𝜑 ∨ 𝜓)))
23		simpl 482	. . . . 5 ⊢ ((𝜑 ∧ 𝜓) → 𝜑)
24		biorf 933	. . . . 5 ⊢ (¬ (𝜑 ∧ 𝜓) → ((𝜑 ⊻ 𝜓) ↔ ((𝜑 ∧ 𝜓) ∨ (𝜑 ⊻ 𝜓))))
25	23, 24	nsyl5 159	. . . 4 ⊢ (¬ 𝜑 → ((𝜑 ⊻ 𝜓) ↔ ((𝜑 ∧ 𝜓) ∨ (𝜑 ⊻ 𝜓))))
26	21, 22, 25	3bitr3d 308	. . 3 ⊢ (¬ 𝜑 → ((𝜑 ∨ 𝜓) ↔ ((𝜑 ∧ 𝜓) ∨ (𝜑 ⊻ 𝜓))))
27	16, 26	pm2.61i 182	. 2 ⊢ ((𝜑 ∨ 𝜓) ↔ ((𝜑 ∧ 𝜓) ∨ (𝜑 ⊻ 𝜓)))
28		3orass 1088	. 2 ⊢ (((𝜑 ∧ 𝜓) ∨ (𝜑 ∧ ¬ 𝜓) ∨ (¬ 𝜑 ∧ 𝜓)) ↔ ((𝜑 ∧ 𝜓) ∨ ((𝜑 ∧ ¬ 𝜓) ∨ (¬ 𝜑 ∧ 𝜓))))
29	2, 27, 28	3bitr4i 302	1 ⊢ ((𝜑 ∨ 𝜓) ↔ ((𝜑 ∧ 𝜓) ∨ (𝜑 ∧ ¬ 𝜓) ∨ (¬ 𝜑 ∧ 𝜓)))

Syntax hints:  ¬ wn 3   → wi 4   ↔ wb 205   ∧ wa 395   ∨ wo 843   ∨ w3o 1084   ⊻ wxo 1503

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8

This theorem depends on definitions:  df-bi 206  df-an 396  df-or 844  df-3or 1086  df-xor 1504

This theorem is referenced by:  uneqsn  41522