Theorem pm4.63dc 871

Description: Theorem *4.63 of [WhiteheadRussell] p. 120, for decidable propositions. (Contributed by Jim Kingdon, 1-May-2018.)

Assertion

Ref	Expression
pm4.63dc	⊢ (DECID 𝜑 → (DECID 𝜓 → (¬ (𝜑 → ¬ 𝜓) ↔ (𝜑 ∧ 𝜓))))

Proof of Theorem pm4.63dc

Step	Hyp	Ref	Expression
1		dfandc 869	. . . 4 ⊢ (DECID 𝜑 → (DECID 𝜓 → ((𝜑 ∧ 𝜓) ↔ ¬ (𝜑 → ¬ 𝜓))))
2	1	imp 123	. . 3 ⊢ ((DECID 𝜑 ∧ DECID 𝜓) → ((𝜑 ∧ 𝜓) ↔ ¬ (𝜑 → ¬ 𝜓)))
3	2	bicomd 140	. 2 ⊢ ((DECID 𝜑 ∧ DECID 𝜓) → (¬ (𝜑 → ¬ 𝜓) ↔ (𝜑 ∧ 𝜓)))
4	3	ex 114	1 ⊢ (DECID 𝜑 → (DECID 𝜓 → (¬ (𝜑 → ¬ 𝜓) ↔ (𝜑 ∧ 𝜓))))

Syntax hints:  ¬ wn 3   → wi 4   ∧ wa 103   ↔ wb 104  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:  pm4.67dc  872