Theorem cbvaldva

Description: Rule used to change the bound variable in a universal quantifier with implicit substitution. Deduction form. Usage of this theorem is discouraged because it depends on ax-13 . Use the weaker cbvaldvaw if possible. (Contributed by David Moews, 1-May-2017) (New usage is discouraged.)

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

Proof

Step	Hyp	Ref	Expression
1		cbvaldva.1	$$⊢\left(\phi \wedge x=y\right)\to \left(\psi \leftrightarrow \chi \right)$$
2		nfv	$$⊢Ⅎy\phi$$
3		nfvd	$$⊢\phi \to Ⅎy\psi$$
4	1	ex	$$⊢\phi \to \left(x=y\to \left(\psi \leftrightarrow \chi \right)\right)$$
5	2 3 4	cbvald	$$⊢\phi \to \left(\forall x\psi \leftrightarrow \forall y\chi \right)$$