Theorem cbvrexdva2

Description: Rule used to change the bound variable in a restricted existential quantifier with implicit substitution which also changes the quantifier domain. Deduction form. (Contributed by David Moews, 1-May-2017) (Proof shortened by Wolf Lammen, 12-Aug-2023)

	Ref	Expression
Hypotheses	cbvraldva2.1	|- ( ( ph /\ x = y ) -> ( ps <-> ch ) )
	cbvraldva2.2	|- ( ( ph /\ x = y ) -> A = B )
Assertion	cbvrexdva2	|- ( ph -> ( E. x e. A ps <-> E. 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	anbi12d	|- ( ( ph /\ x = y ) -> ( ( x e. A /\ ps ) <-> ( y e. B /\ ch ) ) )
6	5	ancoms	|- ( ( x = y /\ ph ) -> ( ( x e. A /\ ps ) <-> ( y e. B /\ ch ) ) )
7	6	pm5.32da	|- ( x = y -> ( ( ph /\ ( x e. A /\ ps ) ) <-> ( ph /\ ( y e. B /\ ch ) ) ) )
8	7	cbvexvw	|- ( E. x ( ph /\ ( x e. A /\ ps ) ) <-> E. y ( ph /\ ( y e. B /\ ch ) ) )
9		19.42v	|- ( E. x ( ph /\ ( x e. A /\ ps ) ) <-> ( ph /\ E. x ( x e. A /\ ps ) ) )
10		19.42v	|- ( E. y ( ph /\ ( y e. B /\ ch ) ) <-> ( ph /\ E. y ( y e. B /\ ch ) ) )
11	8 9 10	3bitr3i	|- ( ( ph /\ E. x ( x e. A /\ ps ) ) <-> ( ph /\ E. y ( y e. B /\ ch ) ) )
12		pm5.32	|- ( ( ph -> ( E. x ( x e. A /\ ps ) <-> E. y ( y e. B /\ ch ) ) ) <-> ( ( ph /\ E. x ( x e. A /\ ps ) ) <-> ( ph /\ E. y ( y e. B /\ ch ) ) ) )
13	11 12	mpbir	|- ( ph -> ( E. x ( x e. A /\ ps ) <-> E. y ( y e. B /\ ch ) ) )
14		df-rex	|- ( E. x e. A ps <-> E. x ( x e. A /\ ps ) )
15		df-rex	|- ( E. y e. B ch <-> E. y ( y e. B /\ ch ) )
16	13 14 15	3bitr4g	|- ( ph -> ( E. x e. A ps <-> E. y e. B ch ) )