Theorem cbvrmovw

Description: Change the bound variable of a restricted at-most-one quantifier using implicit substitution. Version of cbvrmov with a disjoint variable condition, which requires fewer axioms. (Contributed by NM, 16-Jun-2017) (Revised by Gino Giotto, 30-Sep-2024)

	Ref	Expression
Hypothesis	cbvralvw.1	$$⊢x=y\to \left(\phi \leftrightarrow \psi \right)$$
Assertion	cbvrmovw	$$⊢{\exists }^{*}x\in A\phi \leftrightarrow {\exists }^{*}y\in A\psi$$

Proof

Step	Hyp	Ref	Expression
1		cbvralvw.1	$$⊢x=y\to \left(\phi \leftrightarrow \psi \right)$$
2		eleq1w	$$⊢x=y\to \left(x\in A\leftrightarrow y\in A\right)$$
3	2 1	anbi12d	$$⊢x=y\to \left(\left(x\in A\wedge \phi \right)\leftrightarrow \left(y\in A\wedge \psi \right)\right)$$
4	3	cbvmovw	$$⊢{\exists }^{*}x\left(x\in A\wedge \phi \right)\leftrightarrow {\exists }^{*}y\left(y\in A\wedge \psi \right)$$
5		df-rmo	$$⊢{\exists }^{*}x\in A\phi \leftrightarrow {\exists }^{*}x\left(x\in A\wedge \phi \right)$$
6		df-rmo	$$⊢{\exists }^{*}y\in A\psi \leftrightarrow {\exists }^{*}y\left(y\in A\wedge \psi \right)$$
7	4 5 6	3bitr4i	$$⊢{\exists }^{*}x\in A\phi \leftrightarrow {\exists }^{*}y\in A\psi$$