Theorem pm3.12dc 865

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 864	. . . 4 ⊢ (DECID 𝜑 → (DECID 𝜓 → (¬ (¬ 𝜑 ∨ ¬ 𝜓) → (𝜑 ∧ 𝜓))))
2	1	imp 115	. . 3 ⊢ ((DECID 𝜑 ∧ DECID 𝜓) → (¬ (¬ 𝜑 ∨ ¬ 𝜓) → (𝜑 ∧ 𝜓)))
3		dcn 746	. . . . . 6 ⊢ (DECID 𝜑 → DECID ¬ 𝜑)
4		dcn 746	. . . . . 6 ⊢ (DECID 𝜓 → DECID ¬ 𝜓)
5		dcor 843	. . . . . 6 ⊢ (DECID ¬ 𝜑 → (DECID ¬ 𝜓 → DECID (¬ 𝜑 ∨ ¬ 𝜓)))
6	3, 4, 5	syl2im 34	. . . . 5 ⊢ (DECID 𝜑 → (DECID 𝜓 → DECID (¬ 𝜑 ∨ ¬ 𝜓)))
7		dfordc 791	. . . . 5 ⊢ (DECID (¬ 𝜑 ∨ ¬ 𝜓) → (((¬ 𝜑 ∨ ¬ 𝜓) ∨ (𝜑 ∧ 𝜓)) ↔ (¬ (¬ 𝜑 ∨ ¬ 𝜓) → (𝜑 ∧ 𝜓))))
8	6, 7	syl6 29	. . . 4 ⊢ (DECID 𝜑 → (DECID 𝜓 → (((¬ 𝜑 ∨ ¬ 𝜓) ∨ (𝜑 ∧ 𝜓)) ↔ (¬ (¬ 𝜑 ∨ ¬ 𝜓) → (𝜑 ∧ 𝜓)))))
9	8	imp 115	. . 3 ⊢ ((DECID 𝜑 ∧ DECID 𝜓) → (((¬ 𝜑 ∨ ¬ 𝜓) ∨ (𝜑 ∧ 𝜓)) ↔ (¬ (¬ 𝜑 ∨ ¬ 𝜓) → (𝜑 ∧ 𝜓))))
10	2, 9	mpbird 156	. 2 ⊢ ((DECID 𝜑 ∧ DECID 𝜓) → ((¬ 𝜑 ∨ ¬ 𝜓) ∨ (𝜑 ∧ 𝜓)))
11	10	ex 108	1 ⊢ (DECID 𝜑 → (DECID 𝜓 → ((¬ 𝜑 ∨ ¬ 𝜓) ∨ (𝜑 ∧ 𝜓))))

Syntax hints:  ¬ wn 3   → wi 4   ∧ wa 97   ↔ wb 98   ∨ wo 629  DECID wdc 742

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 99  ax-ia2 100  ax-ia3 101  ax-in1 544  ax-in2 545  ax-io 630

This theorem depends on definitions:  df-bi 110  df-dc 743

This theorem is referenced by: (None)