Theorem dn1dc 950

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 𝜒 ∧ DECID 𝜃))) → (¬ (¬ (¬ (𝜑 ∨ 𝜓) ∨ 𝜒) ∨ ¬ (𝜑 ∨ ¬ (¬ 𝜒 ∨ ¬ (𝜒 ∨ 𝜃)))) ↔ 𝜒))

Proof of Theorem dn1dc

Step	Hyp	Ref	Expression
1		pm2.45 728	. . . . 5 ⊢ (¬ (𝜑 ∨ 𝜓) → ¬ 𝜑)
2		imnan 680	. . . . 5 ⊢ ((¬ (𝜑 ∨ 𝜓) → ¬ 𝜑) ↔ ¬ (¬ (𝜑 ∨ 𝜓) ∧ 𝜑))
3	1, 2	mpbi 144	. . . 4 ⊢ ¬ (¬ (𝜑 ∨ 𝜓) ∧ 𝜑)
4	3	biorfi 736	. . 3 ⊢ (𝜒 ↔ (𝜒 ∨ (¬ (𝜑 ∨ 𝜓) ∧ 𝜑)))
5		orcom 718	. . 3 ⊢ ((𝜒 ∨ (¬ (𝜑 ∨ 𝜓) ∧ 𝜑)) ↔ ((¬ (𝜑 ∨ 𝜓) ∧ 𝜑) ∨ 𝜒))
6		ordir 807	. . 3 ⊢ (((¬ (𝜑 ∨ 𝜓) ∧ 𝜑) ∨ 𝜒) ↔ ((¬ (𝜑 ∨ 𝜓) ∨ 𝜒) ∧ (𝜑 ∨ 𝜒)))
7	4, 5, 6	3bitri 205	. 2 ⊢ (𝜒 ↔ ((¬ (𝜑 ∨ 𝜓) ∨ 𝜒) ∧ (𝜑 ∨ 𝜒)))
8		pm4.45 774	. . . . . 6 ⊢ (𝜒 ↔ (𝜒 ∧ (𝜒 ∨ 𝜃)))
9		simprrl 529	. . . . . . 7 ⊢ ((DECID 𝜑 ∧ (DECID 𝜓 ∧ (DECID 𝜒 ∧ DECID 𝜃))) → DECID 𝜒)
10		dcor 925	. . . . . . . . 9 ⊢ (DECID 𝜒 → (DECID 𝜃 → DECID (𝜒 ∨ 𝜃)))
11	10	imp 123	. . . . . . . 8 ⊢ ((DECID 𝜒 ∧ DECID 𝜃) → DECID (𝜒 ∨ 𝜃))
12	11	ad2antll 483	. . . . . . 7 ⊢ ((DECID 𝜑 ∧ (DECID 𝜓 ∧ (DECID 𝜒 ∧ DECID 𝜃))) → DECID (𝜒 ∨ 𝜃))
13		anordc 946	. . . . . . 7 ⊢ (DECID 𝜒 → (DECID (𝜒 ∨ 𝜃) → ((𝜒 ∧ (𝜒 ∨ 𝜃)) ↔ ¬ (¬ 𝜒 ∨ ¬ (𝜒 ∨ 𝜃)))))
14	9, 12, 13	sylc 62	. . . . . 6 ⊢ ((DECID 𝜑 ∧ (DECID 𝜓 ∧ (DECID 𝜒 ∧ DECID 𝜃))) → ((𝜒 ∧ (𝜒 ∨ 𝜃)) ↔ ¬ (¬ 𝜒 ∨ ¬ (𝜒 ∨ 𝜃))))
15	8, 14	syl5bb 191	. . . . 5 ⊢ ((DECID 𝜑 ∧ (DECID 𝜓 ∧ (DECID 𝜒 ∧ DECID 𝜃))) → (𝜒 ↔ ¬ (¬ 𝜒 ∨ ¬ (𝜒 ∨ 𝜃))))
16	15	orbi2d 780	. . . 4 ⊢ ((DECID 𝜑 ∧ (DECID 𝜓 ∧ (DECID 𝜒 ∧ DECID 𝜃))) → ((𝜑 ∨ 𝜒) ↔ (𝜑 ∨ ¬ (¬ 𝜒 ∨ ¬ (𝜒 ∨ 𝜃)))))
17	16	anbi2d 460	. . 3 ⊢ ((DECID 𝜑 ∧ (DECID 𝜓 ∧ (DECID 𝜒 ∧ DECID 𝜃))) → (((¬ (𝜑 ∨ 𝜓) ∨ 𝜒) ∧ (𝜑 ∨ 𝜒)) ↔ ((¬ (𝜑 ∨ 𝜓) ∨ 𝜒) ∧ (𝜑 ∨ ¬ (¬ 𝜒 ∨ ¬ (𝜒 ∨ 𝜃))))))
18		dcor 925	. . . . . . . 8 ⊢ (DECID 𝜑 → (DECID 𝜓 → DECID (𝜑 ∨ 𝜓)))
19		dcn 832	. . . . . . . 8 ⊢ (DECID (𝜑 ∨ 𝜓) → DECID ¬ (𝜑 ∨ 𝜓))
20	18, 19	syl6 33	. . . . . . 7 ⊢ (DECID 𝜑 → (DECID 𝜓 → DECID ¬ (𝜑 ∨ 𝜓)))
21	20	imp 123	. . . . . 6 ⊢ ((DECID 𝜑 ∧ DECID 𝜓) → DECID ¬ (𝜑 ∨ 𝜓))
22	21	adantrr 471	. . . . 5 ⊢ ((DECID 𝜑 ∧ (DECID 𝜓 ∧ (DECID 𝜒 ∧ DECID 𝜃))) → DECID ¬ (𝜑 ∨ 𝜓))
23		dcor 925	. . . . 5 ⊢ (DECID ¬ (𝜑 ∨ 𝜓) → (DECID 𝜒 → DECID (¬ (𝜑 ∨ 𝜓) ∨ 𝜒)))
24	22, 9, 23	sylc 62	. . . 4 ⊢ ((DECID 𝜑 ∧ (DECID 𝜓 ∧ (DECID 𝜒 ∧ DECID 𝜃))) → DECID (¬ (𝜑 ∨ 𝜓) ∨ 𝜒))
25		dcn 832	. . . . . . . 8 ⊢ (DECID 𝜒 → DECID ¬ 𝜒)
26	9, 25	syl 14	. . . . . . 7 ⊢ ((DECID 𝜑 ∧ (DECID 𝜓 ∧ (DECID 𝜒 ∧ DECID 𝜃))) → DECID ¬ 𝜒)
27		dcn 832	. . . . . . . 8 ⊢ (DECID (𝜒 ∨ 𝜃) → DECID ¬ (𝜒 ∨ 𝜃))
28	12, 27	syl 14	. . . . . . 7 ⊢ ((DECID 𝜑 ∧ (DECID 𝜓 ∧ (DECID 𝜒 ∧ DECID 𝜃))) → DECID ¬ (𝜒 ∨ 𝜃))
29		dcor 925	. . . . . . 7 ⊢ (DECID ¬ 𝜒 → (DECID ¬ (𝜒 ∨ 𝜃) → DECID (¬ 𝜒 ∨ ¬ (𝜒 ∨ 𝜃))))
30	26, 28, 29	sylc 62	. . . . . 6 ⊢ ((DECID 𝜑 ∧ (DECID 𝜓 ∧ (DECID 𝜒 ∧ DECID 𝜃))) → DECID (¬ 𝜒 ∨ ¬ (𝜒 ∨ 𝜃)))
31		dcn 832	. . . . . 6 ⊢ (DECID (¬ 𝜒 ∨ ¬ (𝜒 ∨ 𝜃)) → DECID ¬ (¬ 𝜒 ∨ ¬ (𝜒 ∨ 𝜃)))
32	30, 31	syl 14	. . . . 5 ⊢ ((DECID 𝜑 ∧ (DECID 𝜓 ∧ (DECID 𝜒 ∧ DECID 𝜃))) → DECID ¬ (¬ 𝜒 ∨ ¬ (𝜒 ∨ 𝜃)))
33		dcor 925	. . . . . 6 ⊢ (DECID 𝜑 → (DECID ¬ (¬ 𝜒 ∨ ¬ (𝜒 ∨ 𝜃)) → DECID (𝜑 ∨ ¬ (¬ 𝜒 ∨ ¬ (𝜒 ∨ 𝜃)))))
34	33	imp 123	. . . . 5 ⊢ ((DECID 𝜑 ∧ DECID ¬ (¬ 𝜒 ∨ ¬ (𝜒 ∨ 𝜃))) → DECID (𝜑 ∨ ¬ (¬ 𝜒 ∨ ¬ (𝜒 ∨ 𝜃))))
35	32, 34	syldan 280	. . . 4 ⊢ ((DECID 𝜑 ∧ (DECID 𝜓 ∧ (DECID 𝜒 ∧ DECID 𝜃))) → DECID (𝜑 ∨ ¬ (¬ 𝜒 ∨ ¬ (𝜒 ∨ 𝜃))))
36		anordc 946	. . . 4 ⊢ (DECID (¬ (𝜑 ∨ 𝜓) ∨ 𝜒) → (DECID (𝜑 ∨ ¬ (¬ 𝜒 ∨ ¬ (𝜒 ∨ 𝜃))) → (((¬ (𝜑 ∨ 𝜓) ∨ 𝜒) ∧ (𝜑 ∨ ¬ (¬ 𝜒 ∨ ¬ (𝜒 ∨ 𝜃)))) ↔ ¬ (¬ (¬ (𝜑 ∨ 𝜓) ∨ 𝜒) ∨ ¬ (𝜑 ∨ ¬ (¬ 𝜒 ∨ ¬ (𝜒 ∨ 𝜃)))))))
37	24, 35, 36	sylc 62	. . 3 ⊢ ((DECID 𝜑 ∧ (DECID 𝜓 ∧ (DECID 𝜒 ∧ DECID 𝜃))) → (((¬ (𝜑 ∨ 𝜓) ∨ 𝜒) ∧ (𝜑 ∨ ¬ (¬ 𝜒 ∨ ¬ (𝜒 ∨ 𝜃)))) ↔ ¬ (¬ (¬ (𝜑 ∨ 𝜓) ∨ 𝜒) ∨ ¬ (𝜑 ∨ ¬ (¬ 𝜒 ∨ ¬ (𝜒 ∨ 𝜃))))))
38	17, 37	bitrd 187	. 2 ⊢ ((DECID 𝜑 ∧ (DECID 𝜓 ∧ (DECID 𝜒 ∧ DECID 𝜃))) → (((¬ (𝜑 ∨ 𝜓) ∨ 𝜒) ∧ (𝜑 ∨ 𝜒)) ↔ ¬ (¬ (¬ (𝜑 ∨ 𝜓) ∨ 𝜒) ∨ ¬ (𝜑 ∨ ¬ (¬ 𝜒 ∨ ¬ (𝜒 ∨ 𝜃))))))
39	7, 38	bitr2id 192	1 ⊢ ((DECID 𝜑 ∧ (DECID 𝜓 ∧ (DECID 𝜒 ∧ DECID 𝜃))) → (¬ (¬ (¬ (𝜑 ∨ 𝜓) ∨ 𝜒) ∨ ¬ (𝜑 ∨ ¬ (¬ 𝜒 ∨ ¬ (𝜒 ∨ 𝜃)))) ↔ 𝜒))

Syntax hints:  ¬ wn 3   → wi 4   ∧ wa 103   ↔ wb 104   ∨ wo 698  DECID wdc 824

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-in1 604  ax-in2 605  ax-io 699

This theorem depends on definitions:  df-bi 116  df-stab 821  df-dc 825

This theorem is referenced by: (None)