Theorem cbvrexdva2

Description: Rule used to change the bound variable in a restricted existential quantifier with implicit substitution which also changes the quantifier domain. Deduction form. (Contributed by David Moews, 1-May-2017) (Proof shortened by Wolf Lammen, 8-Jan-2025)

	Ref	Expression
Hypotheses	cbvraldva2.1	$$⊢\left(\phi \wedge x=y\right)\to \left(\psi \leftrightarrow \chi \right)$$
	cbvraldva2.2	$$⊢\left(\phi \wedge x=y\right)\to A=B$$
Assertion	cbvrexdva2	$$⊢\phi \to \left(\exists x\in A\psi \leftrightarrow \exists y\in B\chi \right)$$

Proof

Step	Hyp	Ref	Expression
1		cbvraldva2.1	$$⊢\left(\phi \wedge x=y\right)\to \left(\psi \leftrightarrow \chi \right)$$
2		cbvraldva2.2	$$⊢\left(\phi \wedge x=y\right)\to A=B$$
3	1	notbid	$$⊢\left(\phi \wedge x=y\right)\to \left(\neg \psi \leftrightarrow \neg \chi \right)$$
4	3 2	cbvraldva2	$$⊢\phi \to \left(\forall x\in A\neg \psi \leftrightarrow \forall y\in B\neg \chi \right)$$
5		ralnex	$$⊢\forall x\in A\neg \psi \leftrightarrow \neg \exists x\in A\psi$$
6		ralnex	$$⊢\forall y\in B\neg \chi \leftrightarrow \neg \exists y\in B\chi$$
7	4 5 6	3bitr3g	$$⊢\phi \to \left(\neg \exists x\in A\psi \leftrightarrow \neg \exists y\in B\chi \right)$$
8	7	con4bid	$$⊢\phi \to \left(\exists x\in A\psi \leftrightarrow \exists y\in B\chi \right)$$