Theorem dfeu

Description: Rederive df-eu from the old definition eu6 . (Contributed by NM, 23-Mar-1995) (Proof shortened by Wolf Lammen, 25-May-2019) (Proof shortened by BJ, 7-Oct-2022) (Proof modification is discouraged.) Use df-eu instead. (New usage is discouraged.)

	Ref	Expression
Assertion	dfeu	$$⊢\exists !x\phi \leftrightarrow \left(\exists x\phi \wedge {\exists }^{*}x\phi \right)$$

Proof

Step	Hyp	Ref	Expression
1		abai	$$⊢\left(\exists x\phi \wedge \exists !x\phi \right)\leftrightarrow \left(\exists x\phi \wedge \left(\exists x\phi \to \exists !x\phi \right)\right)$$
2		euex	$$⊢\exists !x\phi \to \exists x\phi$$
3	2	pm4.71ri	$$⊢\exists !x\phi \leftrightarrow \left(\exists x\phi \wedge \exists !x\phi \right)$$
4		moeu	$$⊢{\exists }^{*}x\phi \leftrightarrow \left(\exists x\phi \to \exists !x\phi \right)$$
5	4	anbi2i	$$⊢\left(\exists x\phi \wedge {\exists }^{*}x\phi \right)\leftrightarrow \left(\exists x\phi \wedge \left(\exists x\phi \to \exists !x\phi \right)\right)$$
6	1 3 5	3bitr4i	$$⊢\exists !x\phi \leftrightarrow \left(\exists x\phi \wedge {\exists }^{*}x\phi \right)$$