Theorem antnestlaw3 35687

Description: A law of nested antecedents. Compare with looinv 203. (Contributed by Adrian Ducourtial, 5-Dec-2025.)

Assertion

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

Proof of Theorem antnestlaw3

Step	Hyp	Ref	Expression
1		antnestlaw3lem 35684	. . 3 ⊢ (¬ (((𝜑 → 𝜒) → 𝜓) → 𝜓) → ¬ (((𝜑 → 𝜓) → 𝜒) → 𝜒))
2	1	con4i 114	. 2 ⊢ ((((𝜑 → 𝜓) → 𝜒) → 𝜒) → (((𝜑 → 𝜒) → 𝜓) → 𝜓))
3		antnestlaw3lem 35684	. . 3 ⊢ (¬ (((𝜑 → 𝜓) → 𝜒) → 𝜒) → ¬ (((𝜑 → 𝜒) → 𝜓) → 𝜓))
4	3	con4i 114	. 2 ⊢ ((((𝜑 → 𝜒) → 𝜓) → 𝜓) → (((𝜑 → 𝜓) → 𝜒) → 𝜒))
5	2, 4	impbii 209	1 ⊢ ((((𝜑 → 𝜓) → 𝜒) → 𝜒) ↔ (((𝜑 → 𝜒) → 𝜓) → 𝜓))

Syntax hints:   → 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