Theorem mof

Description: Version of df-mo with disjoint variable condition replaced by nonfreeness hypothesis. (Contributed by NM, 8-Mar-1995) Extract dfmo from this proof, and prove mof from it (as of 30-Sep-2022, directly from df-mo ). (Revised by Wolf Lammen, 28-May-2019) Avoid ax-13 . (Revised by Wolf Lammen, 16-Oct-2022)

	Ref	Expression
Hypothesis	mof.1	$$⊢Ⅎy\phi$$
Assertion	mof	$$⊢{\exists }^{*}x\phi \leftrightarrow \exists y\forall x\left(\phi \to x=y\right)$$

Proof

Step	Hyp	Ref	Expression
1		mof.1	$$⊢Ⅎy\phi$$
2		dfmo	$$⊢{\exists }^{*}x\phi \leftrightarrow \exists z\forall x\left(\phi \to x=z\right)$$
3		nfv	$$⊢Ⅎyx=z$$
4	1 3	nfim	$$⊢Ⅎy\left(\phi \to x=z\right)$$
5	4	nfal	$$⊢Ⅎy\forall x\left(\phi \to x=z\right)$$
6		nfv	$$⊢Ⅎz\forall x\left(\phi \to x=y\right)$$
7		equequ2	$$⊢z=y\to \left(x=z\leftrightarrow x=y\right)$$
8	7	imbi2d	$$⊢z=y\to \left(\left(\phi \to x=z\right)\leftrightarrow \left(\phi \to x=y\right)\right)$$
9	8	albidv	$$⊢z=y\to \left(\forall x\left(\phi \to x=z\right)\leftrightarrow \forall x\left(\phi \to x=y\right)\right)$$
10	5 6 9	cbvexv1	$$⊢\exists z\forall x\left(\phi \to x=z\right)\leftrightarrow \exists y\forall x\left(\phi \to x=y\right)$$
11	2 10	bitri	$$⊢{\exists }^{*}x\phi \leftrightarrow \exists y\forall x\left(\phi \to x=y\right)$$