Theorem cbval

Description: Rule used to change bound variables, using implicit substitution. Usage of this theorem is discouraged because it depends on ax-13 . Check out cbvalw , cbvalvw , cbvalv1 for versions requiring fewer axioms. (Contributed by NM, 13-May-1993) (Revised by Mario Carneiro, 3-Oct-2016) (New usage is discouraged.)

	Ref	Expression
Hypotheses	cbval.1	$$⊢Ⅎy\phi$$
	cbval.2	$$⊢Ⅎx\psi$$
	cbval.3	$$⊢x=y\to \left(\phi \leftrightarrow \psi \right)$$
Assertion	cbval	$$⊢\forall x\phi \leftrightarrow \forall y\psi$$

Proof

Step	Hyp	Ref	Expression
1		cbval.1	$$⊢Ⅎy\phi$$
2		cbval.2	$$⊢Ⅎx\psi$$
3		cbval.3	$$⊢x=y\to \left(\phi \leftrightarrow \psi \right)$$
4	3	biimpd	$$⊢x=y\to \left(\phi \to \psi \right)$$
5	1 2 4	cbv3	$$⊢\forall x\phi \to \forall y\psi$$
6	3	biimprd	$$⊢x=y\to \left(\psi \to \phi \right)$$
7	6	equcoms	$$⊢y=x\to \left(\psi \to \phi \right)$$
8	2 1 7	cbv3	$$⊢\forall y\psi \to \forall x\phi$$
9	5 8	impbii	$$⊢\forall x\phi \leftrightarrow \forall y\psi$$