Users' Mathboxes Mathbox for Richard Penner < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >   Mathboxes  >  or3or Structured version   Visualization version   GIF version

Theorem or3or 37798
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 1466 . . 3 ((𝜑𝜓) ↔ ((𝜑 ∧ ¬ 𝜓) ∨ (¬ 𝜑𝜓)))
21orbi2i 541 . 2 (((𝜑𝜓) ∨ (𝜑𝜓)) ↔ ((𝜑𝜓) ∨ ((𝜑 ∧ ¬ 𝜓) ∨ (¬ 𝜑𝜓))))
3 orc 400 . . . 4 (𝜑 → (𝜑𝜓))
4 exmid 431 . . . . 5 (𝜓 ∨ ¬ 𝜓)
5 pm3.2 463 . . . . . 6 (𝜑 → (𝜓 → (𝜑𝜓)))
6 biimp 205 . . . . . . . . . 10 ((𝜑𝜓) → (𝜑𝜓))
7 iman 440 . . . . . . . . . 10 ((𝜑𝜓) ↔ ¬ (𝜑 ∧ ¬ 𝜓))
86, 7sylib 208 . . . . . . . . 9 ((𝜑𝜓) → ¬ (𝜑 ∧ ¬ 𝜓))
98con2i 134 . . . . . . . 8 ((𝜑 ∧ ¬ 𝜓) → ¬ (𝜑𝜓))
109ex 450 . . . . . . 7 (𝜑 → (¬ 𝜓 → ¬ (𝜑𝜓)))
11 df-xor 1462 . . . . . . . 8 ((𝜑𝜓) ↔ ¬ (𝜑𝜓))
1211bicomi 214 . . . . . . 7 (¬ (𝜑𝜓) ↔ (𝜑𝜓))
1310, 12syl6ib 241 . . . . . 6 (𝜑 → (¬ 𝜓 → (𝜑𝜓)))
145, 13orim12d 882 . . . . 5 (𝜑 → ((𝜓 ∨ ¬ 𝜓) → ((𝜑𝜓) ∨ (𝜑𝜓))))
154, 14mpi 20 . . . 4 (𝜑 → ((𝜑𝜓) ∨ (𝜑𝜓)))
163, 152thd 255 . . 3 (𝜑 → ((𝜑𝜓) ↔ ((𝜑𝜓) ∨ (𝜑𝜓))))
17 bicom 212 . . . . . . 7 ((𝜑𝜓) ↔ (𝜓𝜑))
18 bibif 361 . . . . . . 7 𝜑 → ((𝜓𝜑) ↔ ¬ 𝜓))
1917, 18syl5bb 272 . . . . . 6 𝜑 → ((𝜑𝜓) ↔ ¬ 𝜓))
2019con2bid 344 . . . . 5 𝜑 → (𝜓 ↔ ¬ (𝜑𝜓)))
2120, 12syl6bb 276 . . . 4 𝜑 → (𝜓 ↔ (𝜑𝜓)))
22 biorf 420 . . . 4 𝜑 → (𝜓 ↔ (𝜑𝜓)))
23 simpl 473 . . . . . 6 ((𝜑𝜓) → 𝜑)
2423con3i 150 . . . . 5 𝜑 → ¬ (𝜑𝜓))
25 biorf 420 . . . . 5 (¬ (𝜑𝜓) → ((𝜑𝜓) ↔ ((𝜑𝜓) ∨ (𝜑𝜓))))
2624, 25syl 17 . . . 4 𝜑 → ((𝜑𝜓) ↔ ((𝜑𝜓) ∨ (𝜑𝜓))))
2721, 22, 263bitr3d 298 . . 3 𝜑 → ((𝜑𝜓) ↔ ((𝜑𝜓) ∨ (𝜑𝜓))))
2816, 27pm2.61i 176 . 2 ((𝜑𝜓) ↔ ((𝜑𝜓) ∨ (𝜑𝜓)))
29 3orass 1039 . 2 (((𝜑𝜓) ∨ (𝜑 ∧ ¬ 𝜓) ∨ (¬ 𝜑𝜓)) ↔ ((𝜑𝜓) ∨ ((𝜑 ∧ ¬ 𝜓) ∨ (¬ 𝜑𝜓))))
302, 28, 293bitr4i 292 1 ((𝜑𝜓) ↔ ((𝜑𝜓) ∨ (𝜑 ∧ ¬ 𝜓) ∨ (¬ 𝜑𝜓)))
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wb 196  wo 383  wa 384  w3o 1035  wxo 1461
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 197  df-or 385  df-an 386  df-3or 1037  df-xor 1462
This theorem is referenced by:  uneqsn  37800
  Copyright terms: Public domain W3C validator