Theorem cbvalvOLD

Description: Obsolete version of cbvalv as of 11-Sep-2023. (Contributed by NM, 5-Aug-1993) Remove dependency on ax-10 . (Revised by Wolf Lammen, 17-Jul-2021) (Proof modification is discouraged.) (New usage is discouraged.)

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

Proof

Step	Hyp	Ref	Expression
1		cbvalv.1	$$⊢x=y\to \left(\phi \leftrightarrow \psi \right)$$
2		ax-5	$$⊢\phi \to \forall y\phi$$
3	2	hbal	$$⊢\forall x\phi \to \forall y\forall x\phi$$
4	1	spv	$$⊢\forall x\phi \to \psi$$
5	3 4	alrimih	$$⊢\forall x\phi \to \forall y\psi$$
6		ax-5	$$⊢\psi \to \forall x\psi$$
7	6	hbal	$$⊢\forall y\psi \to \forall x\forall y\psi$$
8	1	equcoms	$$⊢y=x\to \left(\phi \leftrightarrow \psi \right)$$
9	8	bicomd	$$⊢y=x\to \left(\psi \leftrightarrow \phi \right)$$
10	9	spv	$$⊢\forall y\psi \to \phi$$
11	7 10	alrimih	$$⊢\forall y\psi \to \forall x\phi$$
12	5 11	impbii	$$⊢\forall x\phi \leftrightarrow \forall y\psi$$