Theorem dn1dc 867

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		pm2.45 657	. . . . 5 ⊢ (¬ (𝜑 ∨ 𝜓) → ¬ 𝜑)
2		imnan 624	. . . . 5 ⊢ ((¬ (𝜑 ∨ 𝜓) → ¬ 𝜑) ↔ ¬ (¬ (𝜑 ∨ 𝜓) ∧ 𝜑))
3	1, 2	mpbi 133	. . . 4 ⊢ ¬ (¬ (𝜑 ∨ 𝜓) ∧ 𝜑)
4	3	biorfi 665	. . 3 ⊢ (𝜒 ↔ (𝜒 ∨ (¬ (𝜑 ∨ 𝜓) ∧ 𝜑)))
5		orcom 647	. . 3 ⊢ ((𝜒 ∨ (¬ (𝜑 ∨ 𝜓) ∧ 𝜑)) ↔ ((¬ (𝜑 ∨ 𝜓) ∧ 𝜑) ∨ 𝜒))
6		ordir 730	. . 3 ⊢ (((¬ (𝜑 ∨ 𝜓) ∧ 𝜑) ∨ 𝜒) ↔ ((¬ (𝜑 ∨ 𝜓) ∨ 𝜒) ∧ (𝜑 ∨ 𝜒)))
7	4, 5, 6	3bitri 195	. 2 ⊢ (𝜒 ↔ ((¬ (𝜑 ∨ 𝜓) ∨ 𝜒) ∧ (𝜑 ∨ 𝜒)))
8		pm4.45 698	. . . . . 6 ⊢ (𝜒 ↔ (𝜒 ∧ (𝜒 ∨ 𝜃)))
9		simprrl 491	. . . . . . 7 ⊢ ((DECID 𝜑 ∧ (DECID 𝜓 ∧ (DECID 𝜒 ∧ DECID 𝜃))) → DECID 𝜒)
10		dcor 843	. . . . . . . . 9 ⊢ (DECID 𝜒 → (DECID 𝜃 → DECID (𝜒 ∨ 𝜃)))
11	10	imp 115	. . . . . . . 8 ⊢ ((DECID 𝜒 ∧ DECID 𝜃) → DECID (𝜒 ∨ 𝜃))
12	11	ad2antll 460	. . . . . . 7 ⊢ ((DECID 𝜑 ∧ (DECID 𝜓 ∧ (DECID 𝜒 ∧ DECID 𝜃))) → DECID (𝜒 ∨ 𝜃))
13		anordc 863	. . . . . . 7 ⊢ (DECID 𝜒 → (DECID (𝜒 ∨ 𝜃) → ((𝜒 ∧ (𝜒 ∨ 𝜃)) ↔ ¬ (¬ 𝜒 ∨ ¬ (𝜒 ∨ 𝜃)))))
14	9, 12, 13	sylc 56	. . . . . 6 ⊢ ((DECID 𝜑 ∧ (DECID 𝜓 ∧ (DECID 𝜒 ∧ DECID 𝜃))) → ((𝜒 ∧ (𝜒 ∨ 𝜃)) ↔ ¬ (¬ 𝜒 ∨ ¬ (𝜒 ∨ 𝜃))))
15	8, 14	syl5bb 181	. . . . 5 ⊢ ((DECID 𝜑 ∧ (DECID 𝜓 ∧ (DECID 𝜒 ∧ DECID 𝜃))) → (𝜒 ↔ ¬ (¬ 𝜒 ∨ ¬ (𝜒 ∨ 𝜃))))
16	15	orbi2d 704	. . . 4 ⊢ ((DECID 𝜑 ∧ (DECID 𝜓 ∧ (DECID 𝜒 ∧ DECID 𝜃))) → ((𝜑 ∨ 𝜒) ↔ (𝜑 ∨ ¬ (¬ 𝜒 ∨ ¬ (𝜒 ∨ 𝜃)))))
17	16	anbi2d 437	. . 3 ⊢ ((DECID 𝜑 ∧ (DECID 𝜓 ∧ (DECID 𝜒 ∧ DECID 𝜃))) → (((¬ (𝜑 ∨ 𝜓) ∨ 𝜒) ∧ (𝜑 ∨ 𝜒)) ↔ ((¬ (𝜑 ∨ 𝜓) ∨ 𝜒) ∧ (𝜑 ∨ ¬ (¬ 𝜒 ∨ ¬ (𝜒 ∨ 𝜃))))))
18		dcor 843	. . . . . . . 8 ⊢ (DECID 𝜑 → (DECID 𝜓 → DECID (𝜑 ∨ 𝜓)))
19		dcn 746	. . . . . . . 8 ⊢ (DECID (𝜑 ∨ 𝜓) → DECID ¬ (𝜑 ∨ 𝜓))
20	18, 19	syl6 29	. . . . . . 7 ⊢ (DECID 𝜑 → (DECID 𝜓 → DECID ¬ (𝜑 ∨ 𝜓)))
21	20	imp 115	. . . . . 6 ⊢ ((DECID 𝜑 ∧ DECID 𝜓) → DECID ¬ (𝜑 ∨ 𝜓))
22	21	adantrr 448	. . . . 5 ⊢ ((DECID 𝜑 ∧ (DECID 𝜓 ∧ (DECID 𝜒 ∧ DECID 𝜃))) → DECID ¬ (𝜑 ∨ 𝜓))
23		dcor 843	. . . . 5 ⊢ (DECID ¬ (𝜑 ∨ 𝜓) → (DECID 𝜒 → DECID (¬ (𝜑 ∨ 𝜓) ∨ 𝜒)))
24	22, 9, 23	sylc 56	. . . 4 ⊢ ((DECID 𝜑 ∧ (DECID 𝜓 ∧ (DECID 𝜒 ∧ DECID 𝜃))) → DECID (¬ (𝜑 ∨ 𝜓) ∨ 𝜒))
25		dcn 746	. . . . . . . 8 ⊢ (DECID 𝜒 → DECID ¬ 𝜒)
26	9, 25	syl 14	. . . . . . 7 ⊢ ((DECID 𝜑 ∧ (DECID 𝜓 ∧ (DECID 𝜒 ∧ DECID 𝜃))) → DECID ¬ 𝜒)
27		dcn 746	. . . . . . . 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 843	. . . . . . 7 ⊢ (DECID ¬ 𝜒 → (DECID ¬ (𝜒 ∨ 𝜃) → DECID (¬ 𝜒 ∨ ¬ (𝜒 ∨ 𝜃))))
30	26, 28, 29	sylc 56	. . . . . 6 ⊢ ((DECID 𝜑 ∧ (DECID 𝜓 ∧ (DECID 𝜒 ∧ DECID 𝜃))) → DECID (¬ 𝜒 ∨ ¬ (𝜒 ∨ 𝜃)))
31		dcn 746	. . . . . 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 843	. . . . . 6 ⊢ (DECID 𝜑 → (DECID ¬ (¬ 𝜒 ∨ ¬ (𝜒 ∨ 𝜃)) → DECID (𝜑 ∨ ¬ (¬ 𝜒 ∨ ¬ (𝜒 ∨ 𝜃)))))
34	33	imp 115	. . . . 5 ⊢ ((DECID 𝜑 ∧ DECID ¬ (¬ 𝜒 ∨ ¬ (𝜒 ∨ 𝜃))) → DECID (𝜑 ∨ ¬ (¬ 𝜒 ∨ ¬ (𝜒 ∨ 𝜃))))
35	32, 34	syldan 266	. . . 4 ⊢ ((DECID 𝜑 ∧ (DECID 𝜓 ∧ (DECID 𝜒 ∧ DECID 𝜃))) → DECID (𝜑 ∨ ¬ (¬ 𝜒 ∨ ¬ (𝜒 ∨ 𝜃))))
36		anordc 863	. . . 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 56	. . 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 177	. 2 ⊢ ((DECID 𝜑 ∧ (DECID 𝜓 ∧ (DECID 𝜒 ∧ DECID 𝜃))) → (((¬ (𝜑 ∨ 𝜓) ∨ 𝜒) ∧ (𝜑 ∨ 𝜒)) ↔ ¬ (¬ (¬ (𝜑 ∨ 𝜓) ∨ 𝜒) ∨ ¬ (𝜑 ∨ ¬ (¬ 𝜒 ∨ ¬ (𝜒 ∨ 𝜃))))))
39	7, 38	syl5rbb 182	1 ⊢ ((DECID 𝜑 ∧ (DECID 𝜓 ∧ (DECID 𝜒 ∧ DECID 𝜃))) → (¬ (¬ (¬ (𝜑 ∨ 𝜓) ∨ 𝜒) ∨ ¬ (𝜑 ∨ ¬ (¬ 𝜒 ∨ ¬ (𝜒 ∨ 𝜃)))) ↔ 𝜒))

Syntax hints:  ¬ wn 3   → wi 4   ∧ wa 97   ↔ wb 98   ∨ wo 629  DECID wdc 742

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 99  ax-ia2 100  ax-ia3 101  ax-in1 544  ax-in2 545  ax-io 630

This theorem depends on definitions:  df-bi 110  df-dc 743

This theorem is referenced by: (None)