Theorem pm4.81dc 847

Description: Theorem *4.81 of [WhiteheadRussell] p. 122, for decidable propositions. This one needs a decidability condition, but compare with pm4.8 655 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 783	. 2 ⊢ (DECID 𝜑 → ((¬ 𝜑 → 𝜑) → 𝜑))
2		pm2.24 583	. 2 ⊢ (𝜑 → (¬ 𝜑 → 𝜑))
3	1, 2	impbid1 140	1 ⊢ (DECID 𝜑 → ((¬ 𝜑 → 𝜑) ↔ 𝜑))

Syntax hints:  ¬ wn 3   → wi 4   ↔ wb 103  DECID wdc 775

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106  ax-in2 577  ax-io 662

This theorem depends on definitions:  df-bi 115  df-dc 776

This theorem is referenced by: (None)