Theorem cbvraldva2

Description: Rule used to change the bound variable in a restricted universal quantifier with implicit substitution which also changes the quantifier domain. Deduction form. (Contributed by David Moews, 1-May-2017)

	Ref	Expression
Hypotheses	cbvraldva2.1	|- ( ( ph /\ x = y ) -> ( ps <-> ch ) )
	cbvraldva2.2	|- ( ( ph /\ x = y ) -> A = B )
Assertion	cbvraldva2	|- ( ph -> ( A. x e. A ps <-> A. y e. B ch ) )

Proof

Step	Hyp	Ref	Expression
1		cbvraldva2.1	|- ( ( ph /\ x = y ) -> ( ps <-> ch ) )
2		cbvraldva2.2	|- ( ( ph /\ x = y ) -> A = B )
3		simpr	|- ( ( ph /\ x = y ) -> x = y )
4	3 2	eleq12d	|- ( ( ph /\ x = y ) -> ( x e. A <-> y e. B ) )
5	4 1	imbi12d	|- ( ( ph /\ x = y ) -> ( ( x e. A -> ps ) <-> ( y e. B -> ch ) ) )
6	5	cbvaldvaw	|- ( ph -> ( A. x ( x e. A -> ps ) <-> A. y ( y e. B -> ch ) ) )
7		df-ral	|- ( A. x e. A ps <-> A. x ( x e. A -> ps ) )
8		df-ral	|- ( A. y e. B ch <-> A. y ( y e. B -> ch ) )
9	6 7 8	3bitr4g	|- ( ph -> ( A. x e. A ps <-> A. y e. B ch ) )