Theorem pm3.12dc 971

Description: Theorem *3.12 of [WhiteheadRussell] p. 111, but for decidable propositions. (Contributed by Jim Kingdon, 22-Apr-2018.)

Assertion

Ref	Expression
pm3.12dc	⊢ (DECID 𝜑 → (DECID 𝜓 → ((¬ 𝜑 ∨ ¬ 𝜓) ∨ (𝜑 ∧ 𝜓))))

Proof of Theorem pm3.12dc

Step	Hyp	Ref	Expression
1		pm3.11dc 970	. . . 4 ⊢ (DECID 𝜑 → (DECID 𝜓 → (¬ (¬ 𝜑 ∨ ¬ 𝜓) → (𝜑 ∧ 𝜓))))
2	1	imp 124	. . 3 ⊢ ((DECID 𝜑 ∧ DECID 𝜓) → (¬ (¬ 𝜑 ∨ ¬ 𝜓) → (𝜑 ∧ 𝜓)))
3		dcn 854	. . . . . 6 ⊢ (DECID 𝜑 → DECID ¬ 𝜑)
4		dcn 854	. . . . . 6 ⊢ (DECID 𝜓 → DECID ¬ 𝜓)
5		dcor 948	. . . . . 6 ⊢ (DECID ¬ 𝜑 → (DECID ¬ 𝜓 → DECID (¬ 𝜑 ∨ ¬ 𝜓)))
6	3, 4, 5	syl2im 38	. . . . 5 ⊢ (DECID 𝜑 → (DECID 𝜓 → DECID (¬ 𝜑 ∨ ¬ 𝜓)))
7		dfordc 904	. . . . 5 ⊢ (DECID (¬ 𝜑 ∨ ¬ 𝜓) → (((¬ 𝜑 ∨ ¬ 𝜓) ∨ (𝜑 ∧ 𝜓)) ↔ (¬ (¬ 𝜑 ∨ ¬ 𝜓) → (𝜑 ∧ 𝜓))))
8	6, 7	syl6 33	. . . 4 ⊢ (DECID 𝜑 → (DECID 𝜓 → (((¬ 𝜑 ∨ ¬ 𝜓) ∨ (𝜑 ∧ 𝜓)) ↔ (¬ (¬ 𝜑 ∨ ¬ 𝜓) → (𝜑 ∧ 𝜓)))))
9	8	imp 124	. . 3 ⊢ ((DECID 𝜑 ∧ DECID 𝜓) → (((¬ 𝜑 ∨ ¬ 𝜓) ∨ (𝜑 ∧ 𝜓)) ↔ (¬ (¬ 𝜑 ∨ ¬ 𝜓) → (𝜑 ∧ 𝜓))))
10	2, 9	mpbird 167	. 2 ⊢ ((DECID 𝜑 ∧ DECID 𝜓) → ((¬ 𝜑 ∨ ¬ 𝜓) ∨ (𝜑 ∧ 𝜓)))
11	10	ex 115	1 ⊢ (DECID 𝜑 → (DECID 𝜓 → ((¬ 𝜑 ∨ ¬ 𝜓) ∨ (𝜑 ∧ 𝜓))))

This proof depends on syntax axioms:  ¬ wn 3   → wi 4   ∧ wa 104   ↔ wb 105   ∨ wo 720  DECID wdc 846

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-in1 623  ax-in2 624  ax-io 721

This proof depends on definitions:  df-bi 117  df-stab 843  df-dc 847

This theorem is used by: (None)