Theorem antnestlaw3lem

Description: Lemma for antnestlaw3 . (Contributed by Adrian Ducourtial, 5-Dec-2025)

	Ref	Expression
Assertion	antnestlaw3lem	⊢ ( ¬ ( ( ( 𝜑 → 𝜓 ) → 𝜒 ) → 𝜒 ) → ¬ ( ( ( 𝜑 → 𝜒 ) → 𝜓 ) → 𝜓 ) )

Proof

Step	Hyp	Ref	Expression
1		conax1	⊢ ( ¬ ( ( ( 𝜑 → 𝜓 ) → 𝜒 ) → 𝜒 ) → ¬ 𝜒 )
2		simplim	⊢ ( ¬ ( ( ( 𝜑 → 𝜓 ) → 𝜒 ) → 𝜒 ) → ( ( 𝜑 → 𝜓 ) → 𝜒 ) )
3	1 2	mtod	⊢ ( ¬ ( ( ( 𝜑 → 𝜓 ) → 𝜒 ) → 𝜒 ) → ¬ ( 𝜑 → 𝜓 ) )
4		simplim	⊢ ( ¬ ( 𝜑 → 𝜓 ) → 𝜑 )
5	3 4	syl	⊢ ( ¬ ( ( ( 𝜑 → 𝜓 ) → 𝜒 ) → 𝜒 ) → 𝜑 )
6	5 1	jcnd	⊢ ( ¬ ( ( ( 𝜑 → 𝜓 ) → 𝜒 ) → 𝜒 ) → ¬ ( 𝜑 → 𝜒 ) )
7	6	pm2.21d	⊢ ( ¬ ( ( ( 𝜑 → 𝜓 ) → 𝜒 ) → 𝜒 ) → ( ( 𝜑 → 𝜒 ) → 𝜓 ) )
8		conax1	⊢ ( ¬ ( 𝜑 → 𝜓 ) → ¬ 𝜓 )
9	3 8	syl	⊢ ( ¬ ( ( ( 𝜑 → 𝜓 ) → 𝜒 ) → 𝜒 ) → ¬ 𝜓 )
10	7 9	jcnd	⊢ ( ¬ ( ( ( 𝜑 → 𝜓 ) → 𝜒 ) → 𝜒 ) → ¬ ( ( ( 𝜑 → 𝜒 ) → 𝜓 ) → 𝜓 ) )