Theorem cbveuwOLD

Description: Obsolete version of cbveuw as of 23-May-2024. (Contributed by NM, 25-Nov-1994) (Revised by Gino Giotto, 10-Jan-2024) (Proof modification is discouraged.) (New usage is discouraged.)

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

Proof

Step	Hyp	Ref	Expression
1		cbveuwOLD.1	$$⊢Ⅎy\phi$$
2		cbveuwOLD.2	$$⊢Ⅎx\psi$$
3		cbveuwOLD.3	$$⊢x=y\to \left(\phi \leftrightarrow \psi \right)$$
4	1	sb8euv	$$⊢\exists !x\phi \leftrightarrow \exists !y\left[y/x\right]\phi$$
5	2 3	sbiev	$$⊢\left[y/x\right]\phi \leftrightarrow \psi$$
6	5	eubii	$$⊢\exists !y\left[y/x\right]\phi \leftrightarrow \exists !y\psi$$
7	4 6	bitri	$$⊢\exists !x\phi \leftrightarrow \exists !y\psi$$