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Theorem dn1dc 878
Description: DN1 for decidable propositions. Without the decidability conditions, DN1 can serve as a single axiom for Boolean algebra. See http://www-unix.mcs.anl.gov/~mccune/papers/basax/v12.pdf. (Contributed by Jim Kingdon, 22-Apr-2018.)
Assertion
Ref Expression
dn1dc ((DECID 𝜑 ∧ (DECID 𝜓 ∧ (DECID 𝜒DECID 𝜃))) → (¬ (¬ (¬ (𝜑𝜓) ∨ 𝜒) ∨ ¬ (𝜑 ∨ ¬ (¬ 𝜒 ∨ ¬ (𝜒𝜃)))) ↔ 𝜒))

Proof of Theorem dn1dc
StepHypRef Expression
1 pm2.45 667 . . . . 5 (¬ (𝜑𝜓) → ¬ 𝜑)
2 imnan 634 . . . . 5 ((¬ (𝜑𝜓) → ¬ 𝜑) ↔ ¬ (¬ (𝜑𝜓) ∧ 𝜑))
31, 2mpbi 137 . . . 4 ¬ (¬ (𝜑𝜓) ∧ 𝜑)
43biorfi 675 . . 3 (𝜒 ↔ (𝜒 ∨ (¬ (𝜑𝜓) ∧ 𝜑)))
5 orcom 657 . . 3 ((𝜒 ∨ (¬ (𝜑𝜓) ∧ 𝜑)) ↔ ((¬ (𝜑𝜓) ∧ 𝜑) ∨ 𝜒))
6 ordir 741 . . 3 (((¬ (𝜑𝜓) ∧ 𝜑) ∨ 𝜒) ↔ ((¬ (𝜑𝜓) ∨ 𝜒) ∧ (𝜑𝜒)))
74, 5, 63bitri 199 . 2 (𝜒 ↔ ((¬ (𝜑𝜓) ∨ 𝜒) ∧ (𝜑𝜒)))
8 pm4.45 708 . . . . . 6 (𝜒 ↔ (𝜒 ∧ (𝜒𝜃)))
9 simprrl 499 . . . . . . 7 ((DECID 𝜑 ∧ (DECID 𝜓 ∧ (DECID 𝜒DECID 𝜃))) → DECID 𝜒)
10 dcor 854 . . . . . . . . 9 (DECID 𝜒 → (DECID 𝜃DECID (𝜒𝜃)))
1110imp 119 . . . . . . . 8 ((DECID 𝜒DECID 𝜃) → DECID (𝜒𝜃))
1211ad2antll 468 . . . . . . 7 ((DECID 𝜑 ∧ (DECID 𝜓 ∧ (DECID 𝜒DECID 𝜃))) → DECID (𝜒𝜃))
13 anordc 874 . . . . . . 7 (DECID 𝜒 → (DECID (𝜒𝜃) → ((𝜒 ∧ (𝜒𝜃)) ↔ ¬ (¬ 𝜒 ∨ ¬ (𝜒𝜃)))))
149, 12, 13sylc 60 . . . . . 6 ((DECID 𝜑 ∧ (DECID 𝜓 ∧ (DECID 𝜒DECID 𝜃))) → ((𝜒 ∧ (𝜒𝜃)) ↔ ¬ (¬ 𝜒 ∨ ¬ (𝜒𝜃))))
158, 14syl5bb 185 . . . . 5 ((DECID 𝜑 ∧ (DECID 𝜓 ∧ (DECID 𝜒DECID 𝜃))) → (𝜒 ↔ ¬ (¬ 𝜒 ∨ ¬ (𝜒𝜃))))
1615orbi2d 714 . . . 4 ((DECID 𝜑 ∧ (DECID 𝜓 ∧ (DECID 𝜒DECID 𝜃))) → ((𝜑𝜒) ↔ (𝜑 ∨ ¬ (¬ 𝜒 ∨ ¬ (𝜒𝜃)))))
1716anbi2d 445 . . 3 ((DECID 𝜑 ∧ (DECID 𝜓 ∧ (DECID 𝜒DECID 𝜃))) → (((¬ (𝜑𝜓) ∨ 𝜒) ∧ (𝜑𝜒)) ↔ ((¬ (𝜑𝜓) ∨ 𝜒) ∧ (𝜑 ∨ ¬ (¬ 𝜒 ∨ ¬ (𝜒𝜃))))))
18 dcor 854 . . . . . . . 8 (DECID 𝜑 → (DECID 𝜓DECID (𝜑𝜓)))
19 dcn 757 . . . . . . . 8 (DECID (𝜑𝜓) → DECID ¬ (𝜑𝜓))
2018, 19syl6 33 . . . . . . 7 (DECID 𝜑 → (DECID 𝜓DECID ¬ (𝜑𝜓)))
2120imp 119 . . . . . 6 ((DECID 𝜑DECID 𝜓) → DECID ¬ (𝜑𝜓))
2221adantrr 456 . . . . 5 ((DECID 𝜑 ∧ (DECID 𝜓 ∧ (DECID 𝜒DECID 𝜃))) → DECID ¬ (𝜑𝜓))
23 dcor 854 . . . . 5 (DECID ¬ (𝜑𝜓) → (DECID 𝜒DECID (¬ (𝜑𝜓) ∨ 𝜒)))
2422, 9, 23sylc 60 . . . 4 ((DECID 𝜑 ∧ (DECID 𝜓 ∧ (DECID 𝜒DECID 𝜃))) → DECID (¬ (𝜑𝜓) ∨ 𝜒))
25 dcn 757 . . . . . . . 8 (DECID 𝜒DECID ¬ 𝜒)
269, 25syl 14 . . . . . . 7 ((DECID 𝜑 ∧ (DECID 𝜓 ∧ (DECID 𝜒DECID 𝜃))) → DECID ¬ 𝜒)
27 dcn 757 . . . . . . . 8 (DECID (𝜒𝜃) → DECID ¬ (𝜒𝜃))
2812, 27syl 14 . . . . . . 7 ((DECID 𝜑 ∧ (DECID 𝜓 ∧ (DECID 𝜒DECID 𝜃))) → DECID ¬ (𝜒𝜃))
29 dcor 854 . . . . . . 7 (DECID ¬ 𝜒 → (DECID ¬ (𝜒𝜃) → DECID𝜒 ∨ ¬ (𝜒𝜃))))
3026, 28, 29sylc 60 . . . . . 6 ((DECID 𝜑 ∧ (DECID 𝜓 ∧ (DECID 𝜒DECID 𝜃))) → DECID𝜒 ∨ ¬ (𝜒𝜃)))
31 dcn 757 . . . . . 6 (DECID𝜒 ∨ ¬ (𝜒𝜃)) → DECID ¬ (¬ 𝜒 ∨ ¬ (𝜒𝜃)))
3230, 31syl 14 . . . . 5 ((DECID 𝜑 ∧ (DECID 𝜓 ∧ (DECID 𝜒DECID 𝜃))) → DECID ¬ (¬ 𝜒 ∨ ¬ (𝜒𝜃)))
33 dcor 854 . . . . . 6 (DECID 𝜑 → (DECID ¬ (¬ 𝜒 ∨ ¬ (𝜒𝜃)) → DECID (𝜑 ∨ ¬ (¬ 𝜒 ∨ ¬ (𝜒𝜃)))))
3433imp 119 . . . . 5 ((DECID 𝜑DECID ¬ (¬ 𝜒 ∨ ¬ (𝜒𝜃))) → DECID (𝜑 ∨ ¬ (¬ 𝜒 ∨ ¬ (𝜒𝜃))))
3532, 34syldan 270 . . . 4 ((DECID 𝜑 ∧ (DECID 𝜓 ∧ (DECID 𝜒DECID 𝜃))) → DECID (𝜑 ∨ ¬ (¬ 𝜒 ∨ ¬ (𝜒𝜃))))
36 anordc 874 . . . 4 (DECID (¬ (𝜑𝜓) ∨ 𝜒) → (DECID (𝜑 ∨ ¬ (¬ 𝜒 ∨ ¬ (𝜒𝜃))) → (((¬ (𝜑𝜓) ∨ 𝜒) ∧ (𝜑 ∨ ¬ (¬ 𝜒 ∨ ¬ (𝜒𝜃)))) ↔ ¬ (¬ (¬ (𝜑𝜓) ∨ 𝜒) ∨ ¬ (𝜑 ∨ ¬ (¬ 𝜒 ∨ ¬ (𝜒𝜃)))))))
3724, 35, 36sylc 60 . . 3 ((DECID 𝜑 ∧ (DECID 𝜓 ∧ (DECID 𝜒DECID 𝜃))) → (((¬ (𝜑𝜓) ∨ 𝜒) ∧ (𝜑 ∨ ¬ (¬ 𝜒 ∨ ¬ (𝜒𝜃)))) ↔ ¬ (¬ (¬ (𝜑𝜓) ∨ 𝜒) ∨ ¬ (𝜑 ∨ ¬ (¬ 𝜒 ∨ ¬ (𝜒𝜃))))))
3817, 37bitrd 181 . 2 ((DECID 𝜑 ∧ (DECID 𝜓 ∧ (DECID 𝜒DECID 𝜃))) → (((¬ (𝜑𝜓) ∨ 𝜒) ∧ (𝜑𝜒)) ↔ ¬ (¬ (¬ (𝜑𝜓) ∨ 𝜒) ∨ ¬ (𝜑 ∨ ¬ (¬ 𝜒 ∨ ¬ (𝜒𝜃))))))
397, 38syl5rbb 186 1 ((DECID 𝜑 ∧ (DECID 𝜓 ∧ (DECID 𝜒DECID 𝜃))) → (¬ (¬ (¬ (𝜑𝜓) ∨ 𝜒) ∨ ¬ (𝜑 ∨ ¬ (¬ 𝜒 ∨ ¬ (𝜒𝜃)))) ↔ 𝜒))
Colors of variables: wff set class
Syntax hints:  ¬ wn 3  wi 4  wa 101  wb 102  wo 639  DECID wdc 753
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 103  ax-ia2 104  ax-ia3 105  ax-in1 554  ax-in2 555  ax-io 640
This theorem depends on definitions:  df-bi 114  df-dc 754
This theorem is referenced by: (None)
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