Metamath Proof Explorer


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, 8-Jan-2025)

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 1 notbid
 |-  ( ( ph /\ x = y ) -> ( -. ps <-> -. ch ) )
4 3 2 cbvraldva2
 |-  ( ph -> ( A. x e. A -. ps <-> A. y e. B -. ch ) )
5 ralnex
 |-  ( A. x e. A -. ps <-> -. E. x e. A ps )
6 ralnex
 |-  ( A. y e. B -. ch <-> -. E. y e. B ch )
7 4 5 6 3bitr3g
 |-  ( ph -> ( -. E. x e. A ps <-> -. E. y e. B ch ) )
8 7 con4bid
 |-  ( ph -> ( E. x e. A ps <-> E. y e. B ch ) )