Theorem bj-pm2.18st 13641

Description: Clavius law for stable formulas. See pm2.18dc 845. (Contributed by BJ, 4-Dec-2023.)

Assertion

Ref	Expression
bj-pm2.18st	⊢ (STAB 𝜑 → ((¬ 𝜑 → 𝜑) → 𝜑))

Proof of Theorem bj-pm2.18st

Step	Hyp	Ref	Expression
1		df-stab 821	. 2 ⊢ (STAB 𝜑 ↔ (¬ ¬ 𝜑 → 𝜑))
2		bj-nnclavius 13628	. . 3 ⊢ ((¬ 𝜑 → 𝜑) → ¬ ¬ 𝜑)
3	2	imim1i 60	. 2 ⊢ ((¬ ¬ 𝜑 → 𝜑) → ((¬ 𝜑 → 𝜑) → 𝜑))
4	1, 3	sylbi 120	1 ⊢ (STAB 𝜑 → ((¬ 𝜑 → 𝜑) → 𝜑))

Syntax hints:  ¬ wn 3   → wi 4  STAB wstab 820

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-in1 604  ax-in2 605

This theorem depends on definitions:  df-bi 116  df-stab 821

This theorem is referenced by: (None)