Theorem pm2.18dc 783

Description: Proof by contradiction for a decidable proposition. Based on Theorem *2.18 of [WhiteheadRussell] p. 103 (also called the Law of Clavius). Intuitionistically it requires a decidability assumption, but compare with pm2.01 578 which does not. (Contributed by Jim Kingdon, 24-Mar-2018.)

Assertion

Ref	Expression
pm2.18dc	⊢ (DECID 𝜑 → ((¬ 𝜑 → 𝜑) → 𝜑))

Proof of Theorem pm2.18dc

Step	Hyp	Ref	Expression
1		pm2.21 579	. . . 4 ⊢ (¬ 𝜑 → (𝜑 → ¬ (¬ 𝜑 → 𝜑)))
2	1	a2i 11	. . 3 ⊢ ((¬ 𝜑 → 𝜑) → (¬ 𝜑 → ¬ (¬ 𝜑 → 𝜑)))
3		condc 782	. . 3 ⊢ (DECID 𝜑 → ((¬ 𝜑 → ¬ (¬ 𝜑 → 𝜑)) → ((¬ 𝜑 → 𝜑) → 𝜑)))
4	2, 3	syl5 32	. 2 ⊢ (DECID 𝜑 → ((¬ 𝜑 → 𝜑) → ((¬ 𝜑 → 𝜑) → 𝜑)))
5	4	pm2.43d 49	1 ⊢ (DECID 𝜑 → ((¬ 𝜑 → 𝜑) → 𝜑))

Syntax hints:  ¬ wn 3   → wi 4  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:  pm4.81dc  847