Theorem wl-dfreuf

Description: Restricted existential uniqueness ( df-wl-reu ) allows a simplification, if we can assume all occurrences of x in A are bounded. (Contributed by Wolf Lammen, 28-May-2023)

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

Proof

Step	Hyp	Ref	Expression
1		wl-dfrexf	$$⊢\underset{\_}{Ⅎ}xA\to \left(\exists x:A\phi \leftrightarrow \exists x\left(x\in A\wedge \phi \right)\right)$$
2		wl-dfrmof	$$⊢\underset{\_}{Ⅎ}xA\to \left({\exists }^{*}x:A\phi \leftrightarrow {\exists }^{*}x\left(x\in A\wedge \phi \right)\right)$$
3	1 2	anbi12d	$$⊢\underset{\_}{Ⅎ}xA\to \left(\left(\exists x:A\phi \wedge {\exists }^{*}x:A\phi \right)\leftrightarrow \left(\exists x\left(x\in A\wedge \phi \right)\wedge {\exists }^{*}x\left(x\in A\wedge \phi \right)\right)\right)$$
4		df-wl-reu	$$⊢\exists !x:A\phi \leftrightarrow \left(\exists x:A\phi \wedge {\exists }^{*}x:A\phi \right)$$
5		df-eu	$$⊢\exists !x\left(x\in A\wedge \phi \right)\leftrightarrow \left(\exists x\left(x\in A\wedge \phi \right)\wedge {\exists }^{*}x\left(x\in A\wedge \phi \right)\right)$$
6	3 4 5	3bitr4g	$$⊢\underset{\_}{Ⅎ}xA\to \left(\exists !x:A\phi \leftrightarrow \exists !x\left(x\in A\wedge \phi \right)\right)$$