Theorem atbiffatnnb

Description: If a implies b, then a implies not not b. (Contributed by Jarvin Udandy, 28-Aug-2016)

	Ref	Expression
Assertion	atbiffatnnb	⊢ ( ( 𝜑 → 𝜓 ) → ( 𝜑 → ¬ ¬ 𝜓 ) )

Proof

Step	Hyp	Ref	Expression
1		idd	⊢ ( 𝜑 → ( 𝜓 → 𝜓 ) )
2		notnotb	⊢ ( 𝜓 ↔ ¬ ¬ 𝜓 )
3	1 2	imbitrdi	⊢ ( 𝜑 → ( 𝜓 → ¬ ¬ 𝜓 ) )
4	3	a2i	⊢ ( ( 𝜑 → 𝜓 ) → ( 𝜑 → ¬ ¬ 𝜓 ) )