Theorem const 853

Description: Contraposition when the antecedent is a negated stable proposition. See comment of condc 854. (Contributed by BJ, 18-Nov-2023.) (Proof shortened by BJ, 11-Nov-2024.)

Assertion

Ref	Expression
const	⊢ (STAB 𝜑 → ((¬ 𝜑 → ¬ 𝜓) → (𝜓 → 𝜑)))

Proof of Theorem const

Step	Hyp	Ref	Expression
1		con2 644	. 2 ⊢ ((¬ 𝜑 → ¬ 𝜓) → (𝜓 → ¬ ¬ 𝜑))
2		df-stab 832	. . 3 ⊢ (STAB 𝜑 ↔ (¬ ¬ 𝜑 → 𝜑))
3	2	biimpi 120	. 2 ⊢ (STAB 𝜑 → (¬ ¬ 𝜑 → 𝜑))
4	1, 3	syl9r 73	1 ⊢ (STAB 𝜑 → ((¬ 𝜑 → ¬ 𝜓) → (𝜓 → 𝜑)))

Syntax hints:  ¬ wn 3   → wi 4  STAB wstab 831

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-in1 615  ax-in2 616

This theorem depends on definitions:  df-bi 117  df-stab 832

This theorem is referenced by:  condc  854