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Theorem or3or 40895
 Description: Decompose disjunction into three cases. (Contributed by RP, 5-Jul-2021.)
Assertion
Ref Expression
or3or ((𝜑𝜓) ↔ ((𝜑𝜓) ∨ (𝜑 ∧ ¬ 𝜓) ∨ (¬ 𝜑𝜓)))

Proof of Theorem or3or
StepHypRef Expression
1 excxor 1508 . . 3 ((𝜑𝜓) ↔ ((𝜑 ∧ ¬ 𝜓) ∨ (¬ 𝜑𝜓)))
21orbi2i 910 . 2 (((𝜑𝜓) ∨ (𝜑𝜓)) ↔ ((𝜑𝜓) ∨ ((𝜑 ∧ ¬ 𝜓) ∨ (¬ 𝜑𝜓))))
3 orc 864 . . . 4 (𝜑 → (𝜑𝜓))
4 exmid 892 . . . . 5 (𝜓 ∨ ¬ 𝜓)
5 pm3.2 473 . . . . . 6 (𝜑 → (𝜓 → (𝜑𝜓)))
6 biimp 218 . . . . . . . . . 10 ((𝜑𝜓) → (𝜑𝜓))
7 iman 405 . . . . . . . . . 10 ((𝜑𝜓) ↔ ¬ (𝜑 ∧ ¬ 𝜓))
86, 7sylib 221 . . . . . . . . 9 ((𝜑𝜓) → ¬ (𝜑 ∧ ¬ 𝜓))
98con2i 141 . . . . . . . 8 ((𝜑 ∧ ¬ 𝜓) → ¬ (𝜑𝜓))
109ex 416 . . . . . . 7 (𝜑 → (¬ 𝜓 → ¬ (𝜑𝜓)))
11 df-xor 1503 . . . . . . . 8 ((𝜑𝜓) ↔ ¬ (𝜑𝜓))
1211bicomi 227 . . . . . . 7 (¬ (𝜑𝜓) ↔ (𝜑𝜓))
1310, 12syl6ib 254 . . . . . 6 (𝜑 → (¬ 𝜓 → (𝜑𝜓)))
145, 13orim12d 962 . . . . 5 (𝜑 → ((𝜓 ∨ ¬ 𝜓) → ((𝜑𝜓) ∨ (𝜑𝜓))))
154, 14mpi 20 . . . 4 (𝜑 → ((𝜑𝜓) ∨ (𝜑𝜓)))
163, 152thd 268 . . 3 (𝜑 → ((𝜑𝜓) ↔ ((𝜑𝜓) ∨ (𝜑𝜓))))
17 bicom 225 . . . . . . 7 ((𝜑𝜓) ↔ (𝜓𝜑))
18 bibif 375 . . . . . . 7 𝜑 → ((𝜓𝜑) ↔ ¬ 𝜓))
1917, 18syl5bb 286 . . . . . 6 𝜑 → ((𝜑𝜓) ↔ ¬ 𝜓))
2019con2bid 358 . . . . 5 𝜑 → (𝜓 ↔ ¬ (𝜑𝜓)))
2120, 12syl6bb 290 . . . 4 𝜑 → (𝜓 ↔ (𝜑𝜓)))
22 biorf 934 . . . 4 𝜑 → (𝜓 ↔ (𝜑𝜓)))
23 simpl 486 . . . . 5 ((𝜑𝜓) → 𝜑)
24 biorf 934 . . . . 5 (¬ (𝜑𝜓) → ((𝜑𝜓) ↔ ((𝜑𝜓) ∨ (𝜑𝜓))))
2523, 24nsyl5 162 . . . 4 𝜑 → ((𝜑𝜓) ↔ ((𝜑𝜓) ∨ (𝜑𝜓))))
2621, 22, 253bitr3d 312 . . 3 𝜑 → ((𝜑𝜓) ↔ ((𝜑𝜓) ∨ (𝜑𝜓))))
2716, 26pm2.61i 185 . 2 ((𝜑𝜓) ↔ ((𝜑𝜓) ∨ (𝜑𝜓)))
28 3orass 1087 . 2 (((𝜑𝜓) ∨ (𝜑 ∧ ¬ 𝜓) ∨ (¬ 𝜑𝜓)) ↔ ((𝜑𝜓) ∨ ((𝜑 ∧ ¬ 𝜓) ∨ (¬ 𝜑𝜓))))
292, 27, 283bitr4i 306 1 ((𝜑𝜓) ↔ ((𝜑𝜓) ∨ (𝜑 ∧ ¬ 𝜓) ∨ (¬ 𝜑𝜓)))
 Colors of variables: wff setvar class Syntax hints:  ¬ wn 3   → wi 4   ↔ wb 209   ∧ wa 399   ∨ wo 844   ∨ w3o 1083   ⊻ wxo 1502 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 210  df-an 400  df-or 845  df-3or 1085  df-xor 1503 This theorem is referenced by:  uneqsn  40897
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