Theorem cbvals

Description: Rule used to change bound variables, using implicit substitution. (Contributed by David A. Wheeler, 12-Jul-2026)

	Ref	Expression
Hypotheses	cbvals.1	$$⊢x=y\to \left(\phi \leftrightarrow \chi \right)$$
	cbvals.2	$$⊢x=y\to \left(\psi \leftrightarrow \theta \right)$$
Assertion	cbvals	$$⊢\forall \exists x\left(\phi \to \psi \right)\leftrightarrow \forall \exists y\left(\chi \to \theta \right)$$

Proof

Step	Hyp	Ref	Expression
1		cbvals.1	$$⊢x=y\to \left(\phi \leftrightarrow \chi \right)$$
2		cbvals.2	$$⊢x=y\to \left(\psi \leftrightarrow \theta \right)$$
3	1 2	imbi12d	$$⊢x=y\to \left(\left(\phi \to \psi \right)\leftrightarrow \left(\chi \to \theta \right)\right)$$
4	3	cbvalvw	$$⊢\forall x\left(\phi \to \psi \right)\leftrightarrow \forall y\left(\chi \to \theta \right)$$
5	1	cbvexvw	$$⊢\exists x\phi \leftrightarrow \exists y\chi$$
6	4 5	anbi12i	$$⊢\left(\forall x\left(\phi \to \psi \right)\wedge \exists x\phi \right)\leftrightarrow \left(\forall y\left(\chi \to \theta \right)\wedge \exists y\chi \right)$$
7		df-als	$$⊢\forall \exists x\left(\phi \to \psi \right)\leftrightarrow \left(\forall x\left(\phi \to \psi \right)\wedge \exists x\phi \right)$$
8		df-als	$$⊢\forall \exists y\left(\chi \to \theta \right)\leftrightarrow \left(\forall y\left(\chi \to \theta \right)\wedge \exists y\chi \right)$$
9	6 7 8	3bitr4i	$$⊢\forall \exists x\left(\phi \to \psi \right)\leftrightarrow \forall \exists y\left(\chi \to \theta \right)$$