Theorem pm4.55dc 933

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 897	. . . . 5 ⊢ (DECID 𝜑 → (DECID 𝜓 → ((¬ 𝜑 ∧ 𝜓) ↔ ¬ (𝜑 ∨ ¬ 𝜓))))
2	1	imp 123	. . . 4 ⊢ ((DECID 𝜑 ∧ DECID 𝜓) → ((¬ 𝜑 ∧ 𝜓) ↔ ¬ (𝜑 ∨ ¬ 𝜓)))
3		dcn 837	. . . . . . . . 9 ⊢ (DECID 𝜓 → DECID ¬ 𝜓)
4	3	anim2i 340	. . . . . . . 8 ⊢ ((DECID 𝜑 ∧ DECID 𝜓) → (DECID 𝜑 ∧ DECID ¬ 𝜓))
5		dcor 930	. . . . . . . . 9 ⊢ (DECID 𝜑 → (DECID ¬ 𝜓 → DECID (𝜑 ∨ ¬ 𝜓)))
6	5	imp 123	. . . . . . . 8 ⊢ ((DECID 𝜑 ∧ DECID ¬ 𝜓) → DECID (𝜑 ∨ ¬ 𝜓))
7	4, 6	syl 14	. . . . . . 7 ⊢ ((DECID 𝜑 ∧ DECID 𝜓) → DECID (𝜑 ∨ ¬ 𝜓))
8		dcn 837	. . . . . . . . 9 ⊢ (DECID 𝜑 → DECID ¬ 𝜑)
9		dcan2 929	. . . . . . . . 9 ⊢ (DECID ¬ 𝜑 → (DECID 𝜓 → DECID (¬ 𝜑 ∧ 𝜓)))
10	8, 9	syl 14	. . . . . . . 8 ⊢ (DECID 𝜑 → (DECID 𝜓 → DECID (¬ 𝜑 ∧ 𝜓)))
11	10	imp 123	. . . . . . 7 ⊢ ((DECID 𝜑 ∧ DECID 𝜓) → DECID (¬ 𝜑 ∧ 𝜓))
12	7, 11	jca 304	. . . . . 6 ⊢ ((DECID 𝜑 ∧ DECID 𝜓) → (DECID (𝜑 ∨ ¬ 𝜓) ∧ DECID (¬ 𝜑 ∧ 𝜓)))
13		con2bidc 870	. . . . . . 7 ⊢ (DECID (𝜑 ∨ ¬ 𝜓) → (DECID (¬ 𝜑 ∧ 𝜓) → (((𝜑 ∨ ¬ 𝜓) ↔ ¬ (¬ 𝜑 ∧ 𝜓)) ↔ ((¬ 𝜑 ∧ 𝜓) ↔ ¬ (𝜑 ∨ ¬ 𝜓)))))
14	13	imp 123	. . . . . 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 157	. . . 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 140	. 2 ⊢ ((DECID 𝜑 ∧ DECID 𝜓) → (¬ (¬ 𝜑 ∧ 𝜓) ↔ (𝜑 ∨ ¬ 𝜓)))
19	18	ex 114	1 ⊢ (DECID 𝜑 → (DECID 𝜓 → (¬ (¬ 𝜑 ∧ 𝜓) ↔ (𝜑 ∨ ¬ 𝜓))))

Syntax hints:  ¬ wn 3   → wi 4   ∧ wa 103   ↔ wb 104   ∨ wo 703  DECID wdc 829

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 609  ax-in2 610  ax-io 704

This theorem depends on definitions:  df-bi 116  df-stab 826  df-dc 830

This theorem is referenced by: (None)