Theorem cbveuw

Description: Version of cbveu with a disjoint variable condition, which does not require ax-10 , ax-13 . (Contributed by NM, 25-Nov-1994) (Revised by Gino Giotto, 23-May-2024)

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

Proof

Step	Hyp	Ref	Expression
1		cbveuw.1	$$⊢Ⅎy\phi$$
2		cbveuw.2	$$⊢Ⅎx\psi$$
3		cbveuw.3	$$⊢x=y\to \left(\phi \leftrightarrow \psi \right)$$
4	1 2 3	cbvexv1	$$⊢\exists x\phi \leftrightarrow \exists y\psi$$
5	1 2 3	cbvmow	$$⊢{\exists }^{*}x\phi \leftrightarrow {\exists }^{*}y\psi$$
6	4 5	anbi12i	$$⊢\left(\exists x\phi \wedge {\exists }^{*}x\phi \right)\leftrightarrow \left(\exists y\psi \wedge {\exists }^{*}y\psi \right)$$
7		df-eu	$$⊢\exists !x\phi \leftrightarrow \left(\exists x\phi \wedge {\exists }^{*}x\phi \right)$$
8		df-eu	$$⊢\exists !y\psi \leftrightarrow \left(\exists y\psi \wedge {\exists }^{*}y\psi \right)$$
9	6 7 8	3bitr4i	$$⊢\exists !x\phi \leftrightarrow \exists !y\psi$$