Theorem pm3.13dc 900

Description: Theorem *3.13 of [WhiteheadRussell] p. 111, but for decidable propositions. The converse, pm3.14 702, holds for all propositions. (Contributed by Jim Kingdon, 22-Apr-2018.)

Assertion

Ref	Expression
pm3.13dc	⊢ (DECID 𝜑 → (DECID 𝜓 → (¬ (𝜑 ∧ 𝜓) → (¬ 𝜑 ∨ ¬ 𝜓))))

Proof of Theorem pm3.13dc

Step	Hyp	Ref	Expression
1		dcn 779	. . 3 ⊢ (DECID 𝜑 → DECID ¬ 𝜑)
2		dcn 779	. . 3 ⊢ (DECID 𝜓 → DECID ¬ 𝜓)
3		dcor 876	. . 3 ⊢ (DECID ¬ 𝜑 → (DECID ¬ 𝜓 → DECID (¬ 𝜑 ∨ ¬ 𝜓)))
4	1, 2, 3	syl2im 38	. 2 ⊢ (DECID 𝜑 → (DECID 𝜓 → DECID (¬ 𝜑 ∨ ¬ 𝜓)))
5		pm3.11dc 898	. 2 ⊢ (DECID 𝜑 → (DECID 𝜓 → (¬ (¬ 𝜑 ∨ ¬ 𝜓) → (𝜑 ∧ 𝜓))))
6		con1dc 786	. 2 ⊢ (DECID (¬ 𝜑 ∨ ¬ 𝜓) → ((¬ (¬ 𝜑 ∨ ¬ 𝜓) → (𝜑 ∧ 𝜓)) → (¬ (𝜑 ∧ 𝜓) → (¬ 𝜑 ∨ ¬ 𝜓))))
7	4, 5, 6	syl6c 65	1 ⊢ (DECID 𝜑 → (DECID 𝜓 → (¬ (𝜑 ∧ 𝜓) → (¬ 𝜑 ∨ ¬ 𝜓))))

Syntax hints:  ¬ wn 3   → wi 4   ∧ wa 102   ∨ wo 661  DECID wdc 775

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106  ax-in1 576  ax-in2 577  ax-io 662

This theorem depends on definitions:  df-bi 115  df-dc 776

This theorem is referenced by: (None)