Theorem pm4.81dc 908

Description: Theorem *4.81 of [WhiteheadRussell] p. 122, for decidable propositions. This one needs a decidability condition, but compare with pm4.8 707 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 855	. 2 ⊢ (DECID 𝜑 → ((¬ 𝜑 → 𝜑) → 𝜑))
2		pm2.24 621	. 2 ⊢ (𝜑 → (¬ 𝜑 → 𝜑))
3	1, 2	impbid1 142	1 ⊢ (DECID 𝜑 → ((¬ 𝜑 → 𝜑) ↔ 𝜑))

Syntax hints:  ¬ wn 3   → wi 4   ↔ wb 105  DECID wdc 834

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-in1 614  ax-in2 615  ax-io 709

This theorem depends on definitions:  df-bi 117  df-stab 831  df-dc 835

This theorem is referenced by: (None)