Theorem pm4.63dc 813

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 811	. . . 4 ⊢ (DECID 𝜑 → (DECID 𝜓 → ((𝜑 ∧ 𝜓) ↔ ¬ (𝜑 → ¬ 𝜓))))
2	1	imp 122	. . 3 ⊢ ((DECID 𝜑 ∧ DECID 𝜓) → ((𝜑 ∧ 𝜓) ↔ ¬ (𝜑 → ¬ 𝜓)))
3	2	bicomd 139	. 2 ⊢ ((DECID 𝜑 ∧ DECID 𝜓) → (¬ (𝜑 → ¬ 𝜓) ↔ (𝜑 ∧ 𝜓)))
4	3	ex 113	1 ⊢ (DECID 𝜑 → (DECID 𝜓 → (¬ (𝜑 → ¬ 𝜓) ↔ (𝜑 ∧ 𝜓))))

Syntax hints:  ¬ wn 3   → wi 4   ∧ wa 102   ↔ wb 103  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:  pm4.67dc  814