Theorem orsird

Description: A lemma for not-or-not elimination, in deduction form. (Contributed by Giovanni Mascellani, 15-Sep-2017)

	Ref	Expression
Hypothesis	orsild.1	$$⊢\phi \to \neg \left(\psi \vee \chi \right)$$
Assertion	orsird	$$⊢\phi \to \neg \chi$$

Proof

Step	Hyp	Ref	Expression
1		orsild.1	$$⊢\phi \to \neg \left(\psi \vee \chi \right)$$
2		ioran	$$⊢\neg \left(\psi \vee \chi \right)\leftrightarrow \left(\neg \psi \wedge \neg \chi \right)$$
3	1 2	sylib	$$⊢\phi \to \left(\neg \psi \wedge \neg \chi \right)$$
4	3	simprd	$$⊢\phi \to \neg \chi$$