Theorem annotanannot

Description: A conjunction with a negated conjunction. (Contributed by AV, 8-Mar-2022) (Proof shortened by Wolf Lammen, 1-Apr-2022)

	Ref	Expression
Assertion	annotanannot	$$⊢\left(\phi \wedge \neg \left(\phi \wedge \psi \right)\right)\leftrightarrow \left(\phi \wedge \neg \psi \right)$$

Proof

Step	Hyp	Ref	Expression
1		ibar	$$⊢\phi \to \left(\psi \leftrightarrow \left(\phi \wedge \psi \right)\right)$$
2	1	bicomd	$$⊢\phi \to \left(\left(\phi \wedge \psi \right)\leftrightarrow \psi \right)$$
3	2	notbid	$$⊢\phi \to \left(\neg \left(\phi \wedge \psi \right)\leftrightarrow \neg \psi \right)$$
4	3	pm5.32i	$$⊢\left(\phi \wedge \neg \left(\phi \wedge \psi \right)\right)\leftrightarrow \left(\phi \wedge \neg \psi \right)$$