Theorem pm2.521dc 764

Description: Theorem *2.521 of [WhiteheadRussell] p. 107, but with an additional decidability condition. (Contributed by Jim Kingdon, 5-May-2018.)

Assertion

Ref	Expression
pm2.521dc	⊢ (DECID 𝜑 → (¬ (𝜑 → 𝜓) → (𝜓 → 𝜑)))

Proof of Theorem pm2.521dc

Step	Hyp	Ref	Expression
1		pm2.52 582	. 2 ⊢ (¬ (𝜑 → 𝜓) → (¬ 𝜑 → ¬ 𝜓))
2		condc 749	. 2 ⊢ (DECID 𝜑 → ((¬ 𝜑 → ¬ 𝜓) → (𝜓 → 𝜑)))
3	1, 2	syl5 28	1 ⊢ (DECID 𝜑 → (¬ (𝜑 → 𝜓) → (𝜓 → 𝜑)))

Syntax hints:  ¬ wn 3   → wi 4  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-in2 545  ax-io 630

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

This theorem is referenced by: (None)