Theorem pm4.81dc 814

Description: Theorem *4.81 of [WhiteheadRussell] p. 122, for decidable propositions. This one needs a decidability condition, but compare with pm4.8 623 which holds for all propositions. (Contributed by Jim Kingdon, 4-Jul-2018.)

Assertion

Ref	Expression
pm4.81dc	⊢ (DECID 𝜑 → ((¬ 𝜑 → 𝜑) ↔ 𝜑))

Proof of Theorem pm4.81dc

Step	Hyp	Ref	Expression
1		pm2.18dc 750	. 2 ⊢ (DECID 𝜑 → ((¬ 𝜑 → 𝜑) → 𝜑))
2		pm2.24 551	. 2 ⊢ (𝜑 → (¬ 𝜑 → 𝜑))
3	1, 2	impbid1 130	1 ⊢ (DECID 𝜑 → ((¬ 𝜑 → 𝜑) ↔ 𝜑))

Syntax hints:  ¬ wn 3   → wi 4   ↔ wb 98  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)