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-10 , ax-13 . (Contributed by NM, 9-Mar-1995) (Revised by GG, 23-May-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		nfv	$$⊢Ⅎyx=z$$
5	1 4	nfim	$$⊢Ⅎy\left(\phi \to x=z\right)$$
6		nfv	$$⊢Ⅎxy=z$$
7	2 6	nfim	$$⊢Ⅎx\left(\psi \to y=z\right)$$
8		equequ1	$$⊢x=y\to \left(x=z\leftrightarrow y=z\right)$$
9	3 8	imbi12d	$$⊢x=y\to \left(\left(\phi \to x=z\right)\leftrightarrow \left(\psi \to y=z\right)\right)$$
10	5 7 9	cbvalv1	$$⊢\forall x\left(\phi \to x=z\right)\leftrightarrow \forall y\left(\psi \to y=z\right)$$
11	10	exbii	$$⊢\exists z\forall x\left(\phi \to x=z\right)\leftrightarrow \exists z\forall y\left(\psi \to y=z\right)$$
12		dfmo	$$⊢{\exists }^{*}x\phi \leftrightarrow \exists z\forall x\left(\phi \to x=z\right)$$
13		dfmo	$$⊢{\exists }^{*}y\psi \leftrightarrow \exists z\forall y\left(\psi \to y=z\right)$$
14	11 12 13	3bitr4i	$$⊢{\exists }^{*}x\phi \leftrightarrow {\exists }^{*}y\psi$$