Theorem pm3.11dc 864

Description: Theorem *3.11 of [WhiteheadRussell] p. 111, but for decidable propositions. The converse, pm3.1 671, holds for all propositions, not just decidable ones. (Contributed by Jim Kingdon, 22-Apr-2018.)

Assertion

Ref	Expression
pm3.11dc	⊢ (DECID 𝜑 → (DECID 𝜓 → (¬ (¬ 𝜑 ∨ ¬ 𝜓) → (𝜑 ∧ 𝜓))))

Proof of Theorem pm3.11dc

Step	Hyp	Ref	Expression
1		anordc 863	. . . 4 ⊢ (DECID 𝜑 → (DECID 𝜓 → ((𝜑 ∧ 𝜓) ↔ ¬ (¬ 𝜑 ∨ ¬ 𝜓))))
2	1	imp 115	. . 3 ⊢ ((DECID 𝜑 ∧ DECID 𝜓) → ((𝜑 ∧ 𝜓) ↔ ¬ (¬ 𝜑 ∨ ¬ 𝜓)))
3	2	biimprd 147	. 2 ⊢ ((DECID 𝜑 ∧ DECID 𝜓) → (¬ (¬ 𝜑 ∨ ¬ 𝜓) → (𝜑 ∧ 𝜓)))
4	3	ex 108	1 ⊢ (DECID 𝜑 → (DECID 𝜓 → (¬ (¬ 𝜑 ∨ ¬ 𝜓) → (𝜑 ∧ 𝜓))))

Syntax hints:  ¬ wn 3   → wi 4   ∧ wa 97   ↔ wb 98   ∨ wo 629  DECID wdc 742

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 99  ax-ia2 100  ax-ia3 101  ax-in1 544  ax-in2 545  ax-io 630

This theorem depends on definitions:  df-bi 110  df-dc 743

This theorem is referenced by:  pm3.12dc  865  pm3.13dc  866