Theorem dfmo2

Description: Rederive df-mo from the old definition moeu . (Contributed by Wolf Lammen, 27-May-2019) (Proof modification is discouraged.) Use dfmo instead. (New usage is discouraged.)

	Ref	Expression
Assertion	dfmo2	$$⊢{\exists }^{*}x\phi \leftrightarrow \exists y\forall x\left(\phi \to x=y\right)$$

Proof

Step	Hyp	Ref	Expression
1		moeu	$$⊢{\exists }^{*}x\phi \leftrightarrow \left(\exists x\phi \to \exists !x\phi \right)$$
2		eu6	$$⊢\exists !x\phi \leftrightarrow \exists y\forall x\left(\phi \leftrightarrow x=y\right)$$
3	2	imbi2i	$$⊢\left(\exists x\phi \to \exists !x\phi \right)\leftrightarrow \left(\exists x\phi \to \exists y\forall x\left(\phi \leftrightarrow x=y\right)\right)$$
4		dfmoeu	$$⊢\left(\exists x\phi \to \exists y\forall x\left(\phi \leftrightarrow x=y\right)\right)\leftrightarrow \exists y\forall x\left(\phi \to x=y\right)$$
5	1 3 4	3bitri	$$⊢{\exists }^{*}x\phi \leftrightarrow \exists y\forall x\left(\phi \to x=y\right)$$