Theorem pm4.81dc 898

Description: Theorem *4.81 of [WhiteheadRussell] p. 122, for decidable propositions. This one needs a decidability condition, but compare with pm4.8 697 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 845	. 2 ⊢ (DECID 𝜑 → ((¬ 𝜑 → 𝜑) → 𝜑))
2		pm2.24 611	. 2 ⊢ (𝜑 → (¬ 𝜑 → 𝜑))
3	1, 2	impbid1 141	1 ⊢ (DECID 𝜑 → ((¬ 𝜑 → 𝜑) ↔ 𝜑))

Syntax hints:  ¬ wn 3   → wi 4   ↔ wb 104  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: (None)