Theorem or3or 40391

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 1507	. . 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		pm3.2 472	. . . . . 6 ⊢ (𝜑 → (𝜓 → (𝜑 ∧ 𝜓)))
6		biimp 217	. . . . . . . . . 10 ⊢ ((𝜑 ↔ 𝜓) → (𝜑 → 𝜓))
7		iman 404	. . . . . . . . . 10 ⊢ ((𝜑 → 𝜓) ↔ ¬ (𝜑 ∧ ¬ 𝜓))
8	6, 7	sylib 220	. . . . . . . . 9 ⊢ ((𝜑 ↔ 𝜓) → ¬ (𝜑 ∧ ¬ 𝜓))
9	8	con2i 141	. . . . . . . 8 ⊢ ((𝜑 ∧ ¬ 𝜓) → ¬ (𝜑 ↔ 𝜓))
10	9	ex 415	. . . . . . 7 ⊢ (𝜑 → (¬ 𝜓 → ¬ (𝜑 ↔ 𝜓)))
11		df-xor 1502	. . . . . . . 8 ⊢ ((𝜑 ⊻ 𝜓) ↔ ¬ (𝜑 ↔ 𝜓))
12	11	bicomi 226	. . . . . . 7 ⊢ (¬ (𝜑 ↔ 𝜓) ↔ (𝜑 ⊻ 𝜓))
13	10, 12	syl6ib 253	. . . . . 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 267	. . 3 ⊢ (𝜑 → ((𝜑 ∨ 𝜓) ↔ ((𝜑 ∧ 𝜓) ∨ (𝜑 ⊻ 𝜓))))
17		bicom 224	. . . . . . 7 ⊢ ((𝜑 ↔ 𝜓) ↔ (𝜓 ↔ 𝜑))
18		bibif 374	. . . . . . 7 ⊢ (¬ 𝜑 → ((𝜓 ↔ 𝜑) ↔ ¬ 𝜓))
19	17, 18	syl5bb 285	. . . . . 6 ⊢ (¬ 𝜑 → ((𝜑 ↔ 𝜓) ↔ ¬ 𝜓))
20	19	con2bid 357	. . . . 5 ⊢ (¬ 𝜑 → (𝜓 ↔ ¬ (𝜑 ↔ 𝜓)))
21	20, 12	syl6bb 289	. . . 4 ⊢ (¬ 𝜑 → (𝜓 ↔ (𝜑 ⊻ 𝜓)))
22		biorf 933	. . . 4 ⊢ (¬ 𝜑 → (𝜓 ↔ (𝜑 ∨ 𝜓)))
23		simpl 485	. . . . . 6 ⊢ ((𝜑 ∧ 𝜓) → 𝜑)
24	23	con3i 157	. . . . 5 ⊢ (¬ 𝜑 → ¬ (𝜑 ∧ 𝜓))
25		biorf 933	. . . . 5 ⊢ (¬ (𝜑 ∧ 𝜓) → ((𝜑 ⊻ 𝜓) ↔ ((𝜑 ∧ 𝜓) ∨ (𝜑 ⊻ 𝜓))))
26	24, 25	syl 17	. . . 4 ⊢ (¬ 𝜑 → ((𝜑 ⊻ 𝜓) ↔ ((𝜑 ∧ 𝜓) ∨ (𝜑 ⊻ 𝜓))))
27	21, 22, 26	3bitr3d 311	. . 3 ⊢ (¬ 𝜑 → ((𝜑 ∨ 𝜓) ↔ ((𝜑 ∧ 𝜓) ∨ (𝜑 ⊻ 𝜓))))
28	16, 27	pm2.61i 184	. 2 ⊢ ((𝜑 ∨ 𝜓) ↔ ((𝜑 ∧ 𝜓) ∨ (𝜑 ⊻ 𝜓)))
29		3orass 1086	. 2 ⊢ (((𝜑 ∧ 𝜓) ∨ (𝜑 ∧ ¬ 𝜓) ∨ (¬ 𝜑 ∧ 𝜓)) ↔ ((𝜑 ∧ 𝜓) ∨ ((𝜑 ∧ ¬ 𝜓) ∨ (¬ 𝜑 ∧ 𝜓))))
30	2, 28, 29	3bitr4i 305	1 ⊢ ((𝜑 ∨ 𝜓) ↔ ((𝜑 ∧ 𝜓) ∨ (𝜑 ∧ ¬ 𝜓) ∨ (¬ 𝜑 ∧ 𝜓)))

Syntax hints:  ¬ wn 3   → wi 4   ↔ wb 208   ∧ wa 398   ∨ wo 843   ∨ w3o 1082   ⊻ wxo 1501

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 209  df-an 399  df-or 844  df-3or 1084  df-xor 1502

This theorem is referenced by:  uneqsn  40393