Theorem cbveu

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 cbveuw , cbveuvw when possible. (Contributed by NM, 25-Nov-1994) (Revised by Mario Carneiro, 7-Oct-2016) (New usage is discouraged.)

	Ref	Expression
Hypotheses	cbveu.1	$$⊢Ⅎy\phi$$
	cbveu.2	$$⊢Ⅎx\psi$$
	cbveu.3	$$⊢x=y\to \left(\phi \leftrightarrow \psi \right)$$
Assertion	cbveu	$$⊢\exists !x\phi \leftrightarrow \exists !y\psi$$

Proof

Step	Hyp	Ref	Expression
1		cbveu.1	$$⊢Ⅎy\phi$$
2		cbveu.2	$$⊢Ⅎx\psi$$
3		cbveu.3	$$⊢x=y\to \left(\phi \leftrightarrow \psi \right)$$
4	1	sb8eu	$$⊢\exists !x\phi \leftrightarrow \exists !y\left[y/x\right]\phi$$
5	2 3	sbie	$$⊢\left[y/x\right]\phi \leftrightarrow \psi$$
6	5	eubii	$$⊢\exists !y\left[y/x\right]\phi \leftrightarrow \exists !y\psi$$
7	4 6	bitri	$$⊢\exists !x\phi \leftrightarrow \exists !y\psi$$