Theorem pm3.13dc 943

Description: Theorem *3.13 of [WhiteheadRussell] p. 111, but for decidable propositions. The converse, pm3.14 742, 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 827	. . 3 ⊢ (DECID 𝜑 → DECID ¬ 𝜑)
2		dcn 827	. . 3 ⊢ (DECID 𝜓 → DECID ¬ 𝜓)
3		dcor 919	. . 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 941	. 2 ⊢ (DECID 𝜑 → (DECID 𝜓 → (¬ (¬ 𝜑 ∨ ¬ 𝜓) → (𝜑 ∧ 𝜓))))
6		con1dc 841	. 2 ⊢ (DECID (¬ 𝜑 ∨ ¬ 𝜓) → ((¬ (¬ 𝜑 ∨ ¬ 𝜓) → (𝜑 ∧ 𝜓)) → (¬ (𝜑 ∧ 𝜓) → (¬ 𝜑 ∨ ¬ 𝜓))))
7	4, 5, 6	syl6c 66	1 ⊢ (DECID 𝜑 → (DECID 𝜓 → (¬ (𝜑 ∧ 𝜓) → (¬ 𝜑 ∨ ¬ 𝜓))))

Syntax hints:  ¬ wn 3   → wi 4   ∧ wa 103   ∨ wo 697  DECID wdc 819

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 603  ax-in2 604  ax-io 698

This theorem depends on definitions:  df-bi 116  df-stab 816  df-dc 820

This theorem is referenced by: (None)