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Theorem or3or 40251
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 1500 . . 3 ((𝜑𝜓) ↔ ((𝜑 ∧ ¬ 𝜓) ∨ (¬ 𝜑𝜓)))
21orbi2i 906 . 2 (((𝜑𝜓) ∨ (𝜑𝜓)) ↔ ((𝜑𝜓) ∨ ((𝜑 ∧ ¬ 𝜓) ∨ (¬ 𝜑𝜓))))
3 orc 861 . . . 4 (𝜑 → (𝜑𝜓))
4 exmid 888 . . . . 5 (𝜓 ∨ ¬ 𝜓)
5 pm3.2 470 . . . . . 6 (𝜑 → (𝜓 → (𝜑𝜓)))
6 biimp 216 . . . . . . . . . 10 ((𝜑𝜓) → (𝜑𝜓))
7 iman 402 . . . . . . . . . 10 ((𝜑𝜓) ↔ ¬ (𝜑 ∧ ¬ 𝜓))
86, 7sylib 219 . . . . . . . . 9 ((𝜑𝜓) → ¬ (𝜑 ∧ ¬ 𝜓))
98con2i 141 . . . . . . . 8 ((𝜑 ∧ ¬ 𝜓) → ¬ (𝜑𝜓))
109ex 413 . . . . . . 7 (𝜑 → (¬ 𝜓 → ¬ (𝜑𝜓)))
11 df-xor 1496 . . . . . . . 8 ((𝜑𝜓) ↔ ¬ (𝜑𝜓))
1211bicomi 225 . . . . . . 7 (¬ (𝜑𝜓) ↔ (𝜑𝜓))
1310, 12syl6ib 252 . . . . . 6 (𝜑 → (¬ 𝜓 → (𝜑𝜓)))
145, 13orim12d 958 . . . . 5 (𝜑 → ((𝜓 ∨ ¬ 𝜓) → ((𝜑𝜓) ∨ (𝜑𝜓))))
154, 14mpi 20 . . . 4 (𝜑 → ((𝜑𝜓) ∨ (𝜑𝜓)))
163, 152thd 266 . . 3 (𝜑 → ((𝜑𝜓) ↔ ((𝜑𝜓) ∨ (𝜑𝜓))))
17 bicom 223 . . . . . . 7 ((𝜑𝜓) ↔ (𝜓𝜑))
18 bibif 373 . . . . . . 7 𝜑 → ((𝜓𝜑) ↔ ¬ 𝜓))
1917, 18syl5bb 284 . . . . . 6 𝜑 → ((𝜑𝜓) ↔ ¬ 𝜓))
2019con2bid 356 . . . . 5 𝜑 → (𝜓 ↔ ¬ (𝜑𝜓)))
2120, 12syl6bb 288 . . . 4 𝜑 → (𝜓 ↔ (𝜑𝜓)))
22 biorf 930 . . . 4 𝜑 → (𝜓 ↔ (𝜑𝜓)))
23 simpl 483 . . . . . 6 ((𝜑𝜓) → 𝜑)
2423con3i 157 . . . . 5 𝜑 → ¬ (𝜑𝜓))
25 biorf 930 . . . . 5 (¬ (𝜑𝜓) → ((𝜑𝜓) ↔ ((𝜑𝜓) ∨ (𝜑𝜓))))
2624, 25syl 17 . . . 4 𝜑 → ((𝜑𝜓) ↔ ((𝜑𝜓) ∨ (𝜑𝜓))))
2721, 22, 263bitr3d 310 . . 3 𝜑 → ((𝜑𝜓) ↔ ((𝜑𝜓) ∨ (𝜑𝜓))))
2816, 27pm2.61i 183 . 2 ((𝜑𝜓) ↔ ((𝜑𝜓) ∨ (𝜑𝜓)))
29 3orass 1082 . 2 (((𝜑𝜓) ∨ (𝜑 ∧ ¬ 𝜓) ∨ (¬ 𝜑𝜓)) ↔ ((𝜑𝜓) ∨ ((𝜑 ∧ ¬ 𝜓) ∨ (¬ 𝜑𝜓))))
302, 28, 293bitr4i 304 1 ((𝜑𝜓) ↔ ((𝜑𝜓) ∨ (𝜑 ∧ ¬ 𝜓) ∨ (¬ 𝜑𝜓)))
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wb 207  wa 396  wo 841  w3o 1078  wxo 1495
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 208  df-an 397  df-or 842  df-3or 1080  df-xor 1496
This theorem is referenced by:  uneqsn  40253
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