Theorem cbvmo

Description: Rule used to change bound variables, using implicit substitution. Usage of this theorem is discouraged because it depends on ax-13 . Use the weaker cbvmow when possible. (Contributed by NM, 9-Mar-1995) (Revised by Andrew Salmon, 8-Jun-2011) (Proof shortened by Wolf Lammen, 4-Jan-2023) (New usage is discouraged.)

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

Proof

Step	Hyp	Ref	Expression
1		cbvmo.1	$$⊢Ⅎy\phi$$
2		cbvmo.2	$$⊢Ⅎx\psi$$
3		cbvmo.3	$$⊢x=y\to \left(\phi \leftrightarrow \psi \right)$$
4	1	sb8mo	$$⊢{\exists }^{*}x\phi \leftrightarrow {\exists }^{*}y\left[y/x\right]\phi$$
5	2 3	sbie	$$⊢\left[y/x\right]\phi \leftrightarrow \psi$$
6	5	mobii	$$⊢{\exists }^{*}y\left[y/x\right]\phi \leftrightarrow {\exists }^{*}y\psi$$
7	4 6	bitri	$$⊢{\exists }^{*}x\phi \leftrightarrow {\exists }^{*}y\psi$$