Theorem cbvraldva

Description: Rule used to change the bound variable in a restricted universal quantifier with implicit substitution. Deduction form. (Contributed by David Moews, 1-May-2017) Avoid ax-9 , ax-ext . (Revised by Wolf Lammen, 9-Mar-2025)

	Ref	Expression
Hypothesis	cbvraldva.1	|- ( ( ph /\ x = y ) -> ( ps <-> ch ) )
Assertion	cbvraldva	|- ( ph -> ( A. x e. A ps <-> A. y e. A ch ) )

Proof

Step	Hyp	Ref	Expression
1		cbvraldva.1	|- ( ( ph /\ x = y ) -> ( ps <-> ch ) )
2	1	ancoms	|- ( ( x = y /\ ph ) -> ( ps <-> ch ) )
3	2	pm5.74da	|- ( x = y -> ( ( ph -> ps ) <-> ( ph -> ch ) ) )
4	3	cbvralvw	|- ( A. x e. A ( ph -> ps ) <-> A. y e. A ( ph -> ch ) )
5		r19.21v	|- ( A. x e. A ( ph -> ps ) <-> ( ph -> A. x e. A ps ) )
6		r19.21v	|- ( A. y e. A ( ph -> ch ) <-> ( ph -> A. y e. A ch ) )
7	4 5 6	3bitr3i	|- ( ( ph -> A. x e. A ps ) <-> ( ph -> A. y e. A ch ) )
8	7	pm5.74ri	|- ( ph -> ( A. x e. A ps <-> A. y e. A ch ) )