Theorem antnestlaw3

Description: A law of nested antecedents. Compare with looinv . (Contributed by Adrian Ducourtial, 5-Dec-2025)

	Ref	Expression
Assertion	antnestlaw3	⊢ ( ( ( ( 𝜑 → 𝜓 ) → 𝜒 ) → 𝜒 ) ↔ ( ( ( 𝜑 → 𝜒 ) → 𝜓 ) → 𝜓 ) )

Proof

Step	Hyp	Ref	Expression
1		antnestlaw3lem	⊢ ( ¬ ( ( ( 𝜑 → 𝜒 ) → 𝜓 ) → 𝜓 ) → ¬ ( ( ( 𝜑 → 𝜓 ) → 𝜒 ) → 𝜒 ) )
2	1	con4i	⊢ ( ( ( ( 𝜑 → 𝜓 ) → 𝜒 ) → 𝜒 ) → ( ( ( 𝜑 → 𝜒 ) → 𝜓 ) → 𝜓 ) )
3		antnestlaw3lem	⊢ ( ¬ ( ( ( 𝜑 → 𝜓 ) → 𝜒 ) → 𝜒 ) → ¬ ( ( ( 𝜑 → 𝜒 ) → 𝜓 ) → 𝜓 ) )
4	3	con4i	⊢ ( ( ( ( 𝜑 → 𝜒 ) → 𝜓 ) → 𝜓 ) → ( ( ( 𝜑 → 𝜓 ) → 𝜒 ) → 𝜒 ) )
5	2 4	impbii	⊢ ( ( ( ( 𝜑 → 𝜓 ) → 𝜒 ) → 𝜒 ) ↔ ( ( ( 𝜑 → 𝜒 ) → 𝜓 ) → 𝜓 ) )