Theorem cbvexdvaw

Description: Rule used to change the bound variable in an existential quantifier with implicit substitution. Deduction form. Version of cbvexdva with a disjoint variable condition, requiring fewer axioms. (Contributed by David Moews, 1-May-2017) (Revised by Gino Giotto, 10-Jan-2024) Reduce axiom usage. (Revised by Wolf Lammen, 10-Feb-2024)

	Ref	Expression
Hypothesis	cbvaldvaw.1	$$⊢\left(\phi \wedge x=y\right)\to \left(\psi \leftrightarrow \chi \right)$$
Assertion	cbvexdvaw	$$⊢\phi \to \left(\exists x\psi \leftrightarrow \exists y\chi \right)$$

Proof

Step	Hyp	Ref	Expression
1		cbvaldvaw.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	cbvaldvaw	$$⊢\phi \to \left(\forall x\neg \psi \leftrightarrow \forall y\neg \chi \right)$$
4		alnex	$$⊢\forall x\neg \psi \leftrightarrow \neg \exists x\psi$$
5		alnex	$$⊢\forall y\neg \chi \leftrightarrow \neg \exists y\chi$$
6	3 4 5	3bitr3g	$$⊢\phi \to \left(\neg \exists x\psi \leftrightarrow \neg \exists y\chi \right)$$
7	6	con4bid	$$⊢\phi \to \left(\exists x\psi \leftrightarrow \exists y\chi \right)$$