Theorem cbvrexdva

Description: Rule used to change the bound variable in a restricted existential quantifier with implicit substitution. Deduction form. (Contributed by David Moews, 1-May-2017)

	Ref	Expression
Hypothesis	cbvraldva.1	$$⊢\left(\phi \wedge x=y\right)\to \left(\psi \leftrightarrow \chi \right)$$
Assertion	cbvrexdva	$$⊢\phi \to \left(\exists x\in A\psi \leftrightarrow \exists y\in A\chi \right)$$

Proof

Step	Hyp	Ref	Expression
1		cbvraldva.1	$$⊢\left(\phi \wedge x=y\right)\to \left(\psi \leftrightarrow \chi \right)$$
2		eqidd	$$⊢\left(\phi \wedge x=y\right)\to A=A$$
3	1 2	cbvrexdva2	$$⊢\phi \to \left(\exists x\in A\psi \leftrightarrow \exists y\in A\chi \right)$$