Theorem antnestlaw1 35685

Description: A law of nested antecedents. The converse direction is a subschema of pm2.27 42. (Contributed by Adrian Ducourtial, 5-Dec-2025.)

Assertion

Ref	Expression
antnestlaw1	⊢ ((((𝜑 → 𝜓) → 𝜓) → 𝜓) ↔ (𝜑 → 𝜓))

Proof of Theorem antnestlaw1

Step	Hyp	Ref	Expression
1		pm2.21 123	. . . 4 ⊢ (¬ (𝜑 → 𝜓) → ((𝜑 → 𝜓) → 𝜓))
2		conax1 170	. . . 4 ⊢ (¬ (𝜑 → 𝜓) → ¬ 𝜓)
3	1, 2	jcnd 163	. . 3 ⊢ (¬ (𝜑 → 𝜓) → ¬ (((𝜑 → 𝜓) → 𝜓) → 𝜓))
4	3	con4i 114	. 2 ⊢ ((((𝜑 → 𝜓) → 𝜓) → 𝜓) → (𝜑 → 𝜓))
5		pm2.27 42	. 2 ⊢ ((𝜑 → 𝜓) → (((𝜑 → 𝜓) → 𝜓) → 𝜓))
6	4, 5	impbii 209	1 ⊢ ((((𝜑 → 𝜓) → 𝜓) → 𝜓) ↔ (𝜑 → 𝜓))

Syntax hints:  ¬ wn 3   → wi 4   ↔ wb 206

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8

This theorem depends on definitions:  df-bi 207

This theorem is referenced by:  antnestALT  35688