Theorem bj-pm2.18st 16778

Description: Clavius law for stable formulas. See pm2.18dc 867. (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 843	. 2 ⊢ (STAB 𝜑 ↔ (¬ ¬ 𝜑 → 𝜑))
2		bj-nnclavius 16765	. . 3 ⊢ ((¬ 𝜑 → 𝜑) → ¬ ¬ 𝜑)
3	2	imim1i 60	. 2 ⊢ ((¬ ¬ 𝜑 → 𝜑) → ((¬ 𝜑 → 𝜑) → 𝜑))
4	1, 3	sylbi 121	1 ⊢ (STAB 𝜑 → ((¬ 𝜑 → 𝜑) → 𝜑))

This proof depends on syntax axioms:  ¬ wn 3   → wi 4  STAB wstab 842

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-in1 623  ax-in2 624

This proof depends on definitions:  df-bi 117  df-stab 843

This theorem is used by: (None)