Theorem antnestlaw1

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

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

Proof

Step	Hyp	Ref	Expression
1		pm2.21	⊢ ( ¬ ( 𝜑 → 𝜓 ) → ( ( 𝜑 → 𝜓 ) → 𝜓 ) )
2		conax1	⊢ ( ¬ ( 𝜑 → 𝜓 ) → ¬ 𝜓 )
3	1 2	jcnd	⊢ ( ¬ ( 𝜑 → 𝜓 ) → ¬ ( ( ( 𝜑 → 𝜓 ) → 𝜓 ) → 𝜓 ) )
4	3	con4i	⊢ ( ( ( ( 𝜑 → 𝜓 ) → 𝜓 ) → 𝜓 ) → ( 𝜑 → 𝜓 ) )
5		pm2.27	⊢ ( ( 𝜑 → 𝜓 ) → ( ( ( 𝜑 → 𝜓 ) → 𝜓 ) → 𝜓 ) )
6	4 5	impbii	⊢ ( ( ( ( 𝜑 → 𝜓 ) → 𝜓 ) → 𝜓 ) ↔ ( 𝜑 → 𝜓 ) )