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) Avoid ax-9 , ax-ext . (Revised by Wolf Lammen, 9-Mar-2025)

	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	1	notbid	$$⊢\left(\phi \wedge x=y\right)\to \left(\neg \psi \leftrightarrow \neg \chi \right)$$
3	2	cbvraldva	$$⊢\phi \to \left(\forall x\in A\neg \psi \leftrightarrow \forall y\in A\neg \chi \right)$$
4		ralnex	$$⊢\forall x\in A\neg \psi \leftrightarrow \neg \exists x\in A\psi$$
5		ralnex	$$⊢\forall y\in A\neg \chi \leftrightarrow \neg \exists y\in A\chi$$
6	3 4 5	3bitr3g	$$⊢\phi \to \left(\neg \exists x\in A\psi \leftrightarrow \neg \exists y\in A\chi \right)$$
7	6	con4bid	$$⊢\phi \to \left(\exists x\in A\psi \leftrightarrow \exists y\in A\chi \right)$$