Theorem cbveuvw

Description: Change bound variable. Uses only Tarski's FOL axiom schemes. See cbveu for a version with fewer disjoint variable conditions but requiring more axioms. (Contributed by NM, 25-Nov-1994) (Revised by Gino Giotto, 30-Sep-2024)

	Ref	Expression
Hypothesis	cbveuvw.1	$$⊢x=y\to \left(\phi \leftrightarrow \psi \right)$$
Assertion	cbveuvw	$$⊢\exists !x\phi \leftrightarrow \exists !y\psi$$

Proof

Step	Hyp	Ref	Expression
1		cbveuvw.1	$$⊢x=y\to \left(\phi \leftrightarrow \psi \right)$$
2	1	cbvexvw	$$⊢\exists x\phi \leftrightarrow \exists y\psi$$
3	1	cbvmovw	$$⊢{\exists }^{*}x\phi \leftrightarrow {\exists }^{*}y\psi$$
4	2 3	anbi12i	$$⊢\left(\exists x\phi \wedge {\exists }^{*}x\phi \right)\leftrightarrow \left(\exists y\psi \wedge {\exists }^{*}y\psi \right)$$
5		df-eu	$$⊢\exists !x\phi \leftrightarrow \left(\exists x\phi \wedge {\exists }^{*}x\phi \right)$$
6		df-eu	$$⊢\exists !y\psi \leftrightarrow \left(\exists y\psi \wedge {\exists }^{*}y\psi \right)$$
7	4 5 6	3bitr4i	$$⊢\exists !x\phi \leftrightarrow \exists !y\psi$$