Theorem const 837

Description: Contraposition of a stable proposition. See comment of condc 838. (Contributed by BJ, 18-Nov-2023.)

Assertion

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

Proof of Theorem const

Step	Hyp	Ref	Expression
1		df-stab 816	. 2 ⊢ (STAB 𝜑 ↔ (¬ ¬ 𝜑 → 𝜑))
2		con3 631	. . 3 ⊢ ((¬ 𝜑 → ¬ 𝜓) → (¬ ¬ 𝜓 → ¬ ¬ 𝜑))
3		notnot 618	. . . 4 ⊢ (𝜓 → ¬ ¬ 𝜓)
4		imim2 55	. . . 4 ⊢ ((¬ ¬ 𝜑 → 𝜑) → ((¬ ¬ 𝜓 → ¬ ¬ 𝜑) → (¬ ¬ 𝜓 → 𝜑)))
5	3, 4	syl7 69	. . 3 ⊢ ((¬ ¬ 𝜑 → 𝜑) → ((¬ ¬ 𝜓 → ¬ ¬ 𝜑) → (𝜓 → 𝜑)))
6	2, 5	syl5 32	. 2 ⊢ ((¬ ¬ 𝜑 → 𝜑) → ((¬ 𝜑 → ¬ 𝜓) → (𝜓 → 𝜑)))
7	1, 6	sylbi 120	1 ⊢ (STAB 𝜑 → ((¬ 𝜑 → ¬ 𝜓) → (𝜓 → 𝜑)))

Syntax hints:  ¬ wn 3   → wi 4  STAB wstab 815

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-in1 603  ax-in2 604

This theorem depends on definitions:  df-bi 116  df-stab 816

This theorem is referenced by:  condc  838