Theorem atbiffatnnb

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

	Ref	Expression
Assertion	atbiffatnnb	$$⊢\left(\phi \to \psi \right)\to \left(\phi \to \neg \neg \psi \right)$$

Proof

Step	Hyp	Ref	Expression
1		idd	$$⊢\phi \to \left(\psi \to \psi \right)$$
2		notnotb	$$⊢\psi \leftrightarrow \neg \neg \psi$$
3	1 2	syl6ib	$$⊢\phi \to \left(\psi \to \neg \neg \psi \right)$$
4	3	a2i	$$⊢\left(\phi \to \psi \right)\to \left(\phi \to \neg \neg \psi \right)$$