Theorem wl-dfreusb

Description: An alternate definition of restricted existential uniqueness ( df-wl-reu ) using substitution. (Contributed by Wolf Lammen, 28-May-2023)

	Ref	Expression
Assertion	wl-dfreusb	$$⊢\exists !x:A\phi \leftrightarrow \exists !y\left(y\in A\wedge \left[y/x\right]\phi \right)$$

Proof

Step	Hyp	Ref	Expression
1		wl-dfrexsb	$$⊢\exists x:A\phi \leftrightarrow \exists y\left(y\in A\wedge \left[y/x\right]\phi \right)$$
2		wl-dfrmosb	$$⊢{\exists }^{*}x:A\phi \leftrightarrow {\exists }^{*}y\left(y\in A\wedge \left[y/x\right]\phi \right)$$
3	1 2	anbi12i	$$⊢\left(\exists x:A\phi \wedge {\exists }^{*}x:A\phi \right)\leftrightarrow \left(\exists y\left(y\in A\wedge \left[y/x\right]\phi \right)\wedge {\exists }^{*}y\left(y\in A\wedge \left[y/x\right]\phi \right)\right)$$
4		df-wl-reu	$$⊢\exists !x:A\phi \leftrightarrow \left(\exists x:A\phi \wedge {\exists }^{*}x:A\phi \right)$$
5		df-eu	$$⊢\exists !y\left(y\in A\wedge \left[y/x\right]\phi \right)\leftrightarrow \left(\exists y\left(y\in A\wedge \left[y/x\right]\phi \right)\wedge {\exists }^{*}y\left(y\in A\wedge \left[y/x\right]\phi \right)\right)$$
6	3 4 5	3bitr4i	$$⊢\exists !x:A\phi \leftrightarrow \exists !y\left(y\in A\wedge \left[y/x\right]\phi \right)$$