Theorem cbveuALT

Description: Alternative proof of cbveu . Since df-eu combines two other quantifiers, one can base this theorem on their associated 'change bounded variable' kind of theorems as well. (Contributed by Wolf Lammen, 5-Jan-2023) (Proof modification is discouraged.) (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	cbveuALT	$$⊢\exists !x\phi \leftrightarrow \exists !y\psi$$

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 2 3	cbvex	$$⊢\exists x\phi \leftrightarrow \exists y\psi$$
5	1 2 3	cbvmo	$$⊢{\exists }^{*}x\phi \leftrightarrow {\exists }^{*}y\psi$$
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	$$⊢\exists !x\phi \leftrightarrow \exists !y\psi$$