Theorem pm4.67dc 888

Description: Theorem *4.67 of [WhiteheadRussell] p. 120, for decidable propositions. (Contributed by Jim Kingdon, 1-May-2018.)

Assertion

Ref	Expression
pm4.67dc	⊢ (DECID 𝜑 → (DECID 𝜓 → (¬ (¬ 𝜑 → ¬ 𝜓) ↔ (¬ 𝜑 ∧ 𝜓))))

Proof of Theorem pm4.67dc

Step	Hyp	Ref	Expression
1		dcn 843	. 2 ⊢ (DECID 𝜑 → DECID ¬ 𝜑)
2		pm4.63dc 887	. 2 ⊢ (DECID ¬ 𝜑 → (DECID 𝜓 → (¬ (¬ 𝜑 → ¬ 𝜓) ↔ (¬ 𝜑 ∧ 𝜓))))
3	1, 2	syl 14	1 ⊢ (DECID 𝜑 → (DECID 𝜓 → (¬ (¬ 𝜑 → ¬ 𝜓) ↔ (¬ 𝜑 ∧ 𝜓))))

Syntax hints:  ¬ wn 3   → wi 4   ∧ wa 104   ↔ wb 105  DECID wdc 835

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-in1 615  ax-in2 616  ax-io 710

This theorem depends on definitions:  df-bi 117  df-stab 832  df-dc 836

This theorem is referenced by: (None)