Theorem cbvmow

Description: Rule used to change bound variables, using implicit substitution. Version of cbvmo with a disjoint variable condition, which does not require ax-13 . (Contributed by NM, 9-Mar-1995) (Revised by Gino Giotto, 10-Jan-2024)

	Ref	Expression
Hypotheses	cbvmow.1	$$⊢Ⅎy\phi$$
	cbvmow.2	$$⊢Ⅎx\psi$$
	cbvmow.3	$$⊢x=y\to \left(\phi \leftrightarrow \psi \right)$$
Assertion	cbvmow	$$⊢{\exists }^{*}x\phi \leftrightarrow {\exists }^{*}y\psi$$

Proof

Step	Hyp	Ref	Expression
1		cbvmow.1	$$⊢Ⅎy\phi$$
2		cbvmow.2	$$⊢Ⅎx\psi$$
3		cbvmow.3	$$⊢x=y\to \left(\phi \leftrightarrow \psi \right)$$
4	1	sb8ev	$$⊢\exists x\phi \leftrightarrow \exists y\left[y/x\right]\phi$$
5	1	sb8euv	$$⊢\exists !x\phi \leftrightarrow \exists !y\left[y/x\right]\phi$$
6	4 5	imbi12i	$$⊢\left(\exists x\phi \to \exists !x\phi \right)\leftrightarrow \left(\exists y\left[y/x\right]\phi \to \exists !y\left[y/x\right]\phi \right)$$
7		moeu	$$⊢{\exists }^{*}x\phi \leftrightarrow \left(\exists x\phi \to \exists !x\phi \right)$$
8		moeu	$$⊢{\exists }^{*}y\left[y/x\right]\phi \leftrightarrow \left(\exists y\left[y/x\right]\phi \to \exists !y\left[y/x\right]\phi \right)$$
9	6 7 8	3bitr4i	$$⊢{\exists }^{*}x\phi \leftrightarrow {\exists }^{*}y\left[y/x\right]\phi$$
10	2 3	sbiev	$$⊢\left[y/x\right]\phi \leftrightarrow \psi$$
11	10	mobii	$$⊢{\exists }^{*}y\left[y/x\right]\phi \leftrightarrow {\exists }^{*}y\psi$$
12	9 11	bitri	$$⊢{\exists }^{*}x\phi \leftrightarrow {\exists }^{*}y\psi$$