Theorem raln

Description: Restricted universally quantified negation expressed as a universally quantified negation. (Contributed by BJ, 16-Jul-2021)

	Ref	Expression
Assertion	raln	$$⊢\forall x\in A\neg \phi \leftrightarrow \forall x\neg \left(x\in A\wedge \phi \right)$$

Proof

Step	Hyp	Ref	Expression
1		df-ral	$$⊢\forall x\in A\neg \phi \leftrightarrow \forall x\left(x\in A\to \neg \phi \right)$$
2		imnang	$$⊢\forall x\left(x\in A\to \neg \phi \right)\leftrightarrow \forall x\neg \left(x\in A\wedge \phi \right)$$
3	1 2	bitri	$$⊢\forall x\in A\neg \phi \leftrightarrow \forall x\neg \left(x\in A\wedge \phi \right)$$