Theorem pm3.12dc 953

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 952	. . . 4 ⊢ (DECID 𝜑 → (DECID 𝜓 → (¬ (¬ 𝜑 ∨ ¬ 𝜓) → (𝜑 ∧ 𝜓))))
2	1	imp 123	. . 3 ⊢ ((DECID 𝜑 ∧ DECID 𝜓) → (¬ (¬ 𝜑 ∨ ¬ 𝜓) → (𝜑 ∧ 𝜓)))
3		dcn 837	. . . . . 6 ⊢ (DECID 𝜑 → DECID ¬ 𝜑)
4		dcn 837	. . . . . 6 ⊢ (DECID 𝜓 → DECID ¬ 𝜓)
5		dcor 930	. . . . . 6 ⊢ (DECID ¬ 𝜑 → (DECID ¬ 𝜓 → DECID (¬ 𝜑 ∨ ¬ 𝜓)))
6	3, 4, 5	syl2im 38	. . . . 5 ⊢ (DECID 𝜑 → (DECID 𝜓 → DECID (¬ 𝜑 ∨ ¬ 𝜓)))
7		dfordc 887	. . . . 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 123	. . 3 ⊢ ((DECID 𝜑 ∧ DECID 𝜓) → (((¬ 𝜑 ∨ ¬ 𝜓) ∨ (𝜑 ∧ 𝜓)) ↔ (¬ (¬ 𝜑 ∨ ¬ 𝜓) → (𝜑 ∧ 𝜓))))
10	2, 9	mpbird 166	. 2 ⊢ ((DECID 𝜑 ∧ DECID 𝜓) → ((¬ 𝜑 ∨ ¬ 𝜓) ∨ (𝜑 ∧ 𝜓)))
11	10	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)