Theorem wl-dfrexv

Description: Alternate definition of restricted universal quantification ( df-wl-rex ) when x and A are disjoint. (Contributed by Wolf Lammen, 25-May-2023)

	Ref	Expression
Assertion	wl-dfrexv	$$⊢\exists x:A\phi \leftrightarrow \exists x\left(x\in A\wedge \phi \right)$$

Proof

Step	Hyp	Ref	Expression
1		wl-dfralv	$$⊢\forall x:A\neg \phi \leftrightarrow \forall x\left(x\in A\to \neg \phi \right)$$
2	1	notbii	$$⊢\neg \forall x:A\neg \phi \leftrightarrow \neg \forall x\left(x\in A\to \neg \phi \right)$$
3		df-wl-rex	$$⊢\exists x:A\phi \leftrightarrow \neg \forall x:A\neg \phi$$
4		exnalimn	$$⊢\exists x\left(x\in A\wedge \phi \right)\leftrightarrow \neg \forall x\left(x\in A\to \neg \phi \right)$$
5	2 3 4	3bitr4i	$$⊢\exists x:A\phi \leftrightarrow \exists x\left(x\in A\wedge \phi \right)$$