Theorem pm2.521gdc 858

Description: A general instance of Theorem *2.521 of [WhiteheadRussell] p. 107, under a decidability condition. (Contributed by BJ, 28-Oct-2023.)

Assertion

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

Proof of Theorem pm2.521gdc

Step	Hyp	Ref	Expression
1		pm2.5gdc 856	. 2 ⊢ (DECID 𝜑 → (¬ (𝜑 → 𝜓) → (¬ 𝜑 → ¬ 𝜒)))
2		condc 843	. 2 ⊢ (DECID 𝜑 → ((¬ 𝜑 → ¬ 𝜒) → (𝜒 → 𝜑)))
3	1, 2	syld 45	1 ⊢ (DECID 𝜑 → (¬ (𝜑 → 𝜓) → (𝜒 → 𝜑)))

Syntax hints:  ¬ wn 3   → wi 4  DECID wdc 824

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 604  ax-in2 605  ax-io 699

This theorem depends on definitions:  df-bi 116  df-stab 821  df-dc 825

This theorem is referenced by:  pm2.521dc  859