Theorem wl-dfrexf

Description: Restricted existential quantification ( df-wl-rex ) allows a simplification, if we can assume all occurrences of x in A are bounded. (Contributed by Wolf Lammen, 25-May-2023)

	Ref	Expression
Assertion	wl-dfrexf	$$⊢\underset{\_}{Ⅎ}xA\to \left(\exists x:A\phi \leftrightarrow \exists x\left(x\in A\wedge \phi \right)\right)$$

Proof

Step	Hyp	Ref	Expression
1		wl-dfralf	$$⊢\underset{\_}{Ⅎ}xA\to \left(\forall x:A\neg \phi \leftrightarrow \forall x\left(x\in A\to \neg \phi \right)\right)$$
2	1	notbid	$$⊢\underset{\_}{Ⅎ}xA\to \left(\neg \forall x:A\neg \phi \leftrightarrow \neg \forall x\left(x\in A\to \neg \phi \right)\right)$$
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	3bitr4g	$$⊢\underset{\_}{Ⅎ}xA\to \left(\exists x:A\phi \leftrightarrow \exists x\left(x\in A\wedge \phi \right)\right)$$