Theorem pm4.55dc 879

Description: Theorem *4.55 of [WhiteheadRussell] p. 120, for decidable propositions. (Contributed by Jim Kingdon, 2-May-2018.)

Assertion

Ref	Expression
pm4.55dc	⊢ (DECID 𝜑 → (DECID 𝜓 → (¬ (¬ 𝜑 ∧ 𝜓) ↔ (𝜑 ∨ ¬ 𝜓))))

Proof of Theorem pm4.55dc

Step	Hyp	Ref	Expression
1		pm4.54dc 838	. . . . 5 ⊢ (DECID 𝜑 → (DECID 𝜓 → ((¬ 𝜑 ∧ 𝜓) ↔ ¬ (𝜑 ∨ ¬ 𝜓))))
2	1	imp 122	. . . 4 ⊢ ((DECID 𝜑 ∧ DECID 𝜓) → ((¬ 𝜑 ∧ 𝜓) ↔ ¬ (𝜑 ∨ ¬ 𝜓)))
3		dcn 779	. . . . . . . . 9 ⊢ (DECID 𝜓 → DECID ¬ 𝜓)
4	3	anim2i 334	. . . . . . . 8 ⊢ ((DECID 𝜑 ∧ DECID 𝜓) → (DECID 𝜑 ∧ DECID ¬ 𝜓))
5		dcor 876	. . . . . . . . 9 ⊢ (DECID 𝜑 → (DECID ¬ 𝜓 → DECID (𝜑 ∨ ¬ 𝜓)))
6	5	imp 122	. . . . . . . 8 ⊢ ((DECID 𝜑 ∧ DECID ¬ 𝜓) → DECID (𝜑 ∨ ¬ 𝜓))
7	4, 6	syl 14	. . . . . . 7 ⊢ ((DECID 𝜑 ∧ DECID 𝜓) → DECID (𝜑 ∨ ¬ 𝜓))
8		dcn 779	. . . . . . . . 9 ⊢ (DECID 𝜑 → DECID ¬ 𝜑)
9		dcan 875	. . . . . . . . 9 ⊢ (DECID ¬ 𝜑 → (DECID 𝜓 → DECID (¬ 𝜑 ∧ 𝜓)))
10	8, 9	syl 14	. . . . . . . 8 ⊢ (DECID 𝜑 → (DECID 𝜓 → DECID (¬ 𝜑 ∧ 𝜓)))
11	10	imp 122	. . . . . . 7 ⊢ ((DECID 𝜑 ∧ DECID 𝜓) → DECID (¬ 𝜑 ∧ 𝜓))
12	7, 11	jca 300	. . . . . 6 ⊢ ((DECID 𝜑 ∧ DECID 𝜓) → (DECID (𝜑 ∨ ¬ 𝜓) ∧ DECID (¬ 𝜑 ∧ 𝜓)))
13		con2bidc 802	. . . . . . 7 ⊢ (DECID (𝜑 ∨ ¬ 𝜓) → (DECID (¬ 𝜑 ∧ 𝜓) → (((𝜑 ∨ ¬ 𝜓) ↔ ¬ (¬ 𝜑 ∧ 𝜓)) ↔ ((¬ 𝜑 ∧ 𝜓) ↔ ¬ (𝜑 ∨ ¬ 𝜓)))))
14	13	imp 122	. . . . . 6 ⊢ ((DECID (𝜑 ∨ ¬ 𝜓) ∧ DECID (¬ 𝜑 ∧ 𝜓)) → (((𝜑 ∨ ¬ 𝜓) ↔ ¬ (¬ 𝜑 ∧ 𝜓)) ↔ ((¬ 𝜑 ∧ 𝜓) ↔ ¬ (𝜑 ∨ ¬ 𝜓))))
15	12, 14	syl 14	. . . . 5 ⊢ ((DECID 𝜑 ∧ DECID 𝜓) → (((𝜑 ∨ ¬ 𝜓) ↔ ¬ (¬ 𝜑 ∧ 𝜓)) ↔ ((¬ 𝜑 ∧ 𝜓) ↔ ¬ (𝜑 ∨ ¬ 𝜓))))
16	15	biimprd 156	. . . 4 ⊢ ((DECID 𝜑 ∧ DECID 𝜓) → (((¬ 𝜑 ∧ 𝜓) ↔ ¬ (𝜑 ∨ ¬ 𝜓)) → ((𝜑 ∨ ¬ 𝜓) ↔ ¬ (¬ 𝜑 ∧ 𝜓))))
17	2, 16	mpd 13	. . 3 ⊢ ((DECID 𝜑 ∧ DECID 𝜓) → ((𝜑 ∨ ¬ 𝜓) ↔ ¬ (¬ 𝜑 ∧ 𝜓)))
18	17	bicomd 139	. 2 ⊢ ((DECID 𝜑 ∧ DECID 𝜓) → (¬ (¬ 𝜑 ∧ 𝜓) ↔ (𝜑 ∨ ¬ 𝜓)))
19	18	ex 113	1 ⊢ (DECID 𝜑 → (DECID 𝜓 → (¬ (¬ 𝜑 ∧ 𝜓) ↔ (𝜑 ∨ ¬ 𝜓))))

Syntax hints:  ¬ wn 3   → wi 4   ∧ wa 102   ↔ wb 103   ∨ wo 661  DECID wdc 775

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106  ax-in1 576  ax-in2 577  ax-io 662

This theorem depends on definitions:  df-bi 115  df-dc 776

This theorem is referenced by: (None)