Theorem pm4.55dc 940

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 903	. . . . 5 ⊢ (DECID 𝜑 → (DECID 𝜓 → ((¬ 𝜑 ∧ 𝜓) ↔ ¬ (𝜑 ∨ ¬ 𝜓))))
2	1	imp 124	. . . 4 ⊢ ((DECID 𝜑 ∧ DECID 𝜓) → ((¬ 𝜑 ∧ 𝜓) ↔ ¬ (𝜑 ∨ ¬ 𝜓)))
3		dcor 937	. . . . . 6 ⊢ (DECID 𝜑 → (DECID ¬ 𝜓 → DECID (𝜑 ∨ ¬ 𝜓)))
4		dcn 843	. . . . . 6 ⊢ (DECID 𝜓 → DECID ¬ 𝜓)
5	3, 4	impel 280	. . . . 5 ⊢ ((DECID 𝜑 ∧ DECID 𝜓) → DECID (𝜑 ∨ ¬ 𝜓))
6		dcn 843	. . . . . 6 ⊢ (DECID 𝜑 → DECID ¬ 𝜑)
7		dcan 935	. . . . . 6 ⊢ ((DECID ¬ 𝜑 ∧ DECID 𝜓) → DECID (¬ 𝜑 ∧ 𝜓))
8	6, 7	sylan 283	. . . . 5 ⊢ ((DECID 𝜑 ∧ DECID 𝜓) → DECID (¬ 𝜑 ∧ 𝜓))
9		con2bidc 876	. . . . 5 ⊢ (DECID (𝜑 ∨ ¬ 𝜓) → (DECID (¬ 𝜑 ∧ 𝜓) → (((𝜑 ∨ ¬ 𝜓) ↔ ¬ (¬ 𝜑 ∧ 𝜓)) ↔ ((¬ 𝜑 ∧ 𝜓) ↔ ¬ (𝜑 ∨ ¬ 𝜓)))))
10	5, 8, 9	sylc 62	. . . 4 ⊢ ((DECID 𝜑 ∧ DECID 𝜓) → (((𝜑 ∨ ¬ 𝜓) ↔ ¬ (¬ 𝜑 ∧ 𝜓)) ↔ ((¬ 𝜑 ∧ 𝜓) ↔ ¬ (𝜑 ∨ ¬ 𝜓))))
11	2, 10	mpbird 167	. . 3 ⊢ ((DECID 𝜑 ∧ DECID 𝜓) → ((𝜑 ∨ ¬ 𝜓) ↔ ¬ (¬ 𝜑 ∧ 𝜓)))
12	11, 11, 11	3bitr2rd 217	. 2 ⊢ ((DECID 𝜑 ∧ DECID 𝜓) → (¬ (¬ 𝜑 ∧ 𝜓) ↔ (𝜑 ∨ ¬ 𝜓)))
13	12	ex 115	1 ⊢ (DECID 𝜑 → (DECID 𝜓 → (¬ (¬ 𝜑 ∧ 𝜓) ↔ (𝜑 ∨ ¬ 𝜓))))

Syntax hints:  ¬ wn 3   → wi 4   ∧ wa 104   ↔ wb 105   ∨ wo 709  DECID wdc 835

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-in1 615  ax-in2 616  ax-io 710

This theorem depends on definitions:  df-bi 117  df-stab 832  df-dc 836

This theorem is referenced by: (None)