This is an inofficial mirror of http://metamath.tirix.org for personal testing of a visualizer extension only.

Metamath Proof Explorer

Theorem cbvrexvw

Description: Change the bound variable of a restricted existential quantifier using implicit substitution. Version of cbvrexv with a disjoint variable condition, which does not require ax-10 , ax-11 , ax-12 , ax-13 . (Contributed by NM, 2-Jun-1998) Avoid ax-13 . (Revised by GG, 10-Jan-2024)

Ref Expression
Hypothesis cbvralvw.1 
|- ( x = y -> ( ph <-> ps ) )
Assertion cbvrexvw 
|- ( E. x e. A ph <-> E. y e. A ps )

Proof

Step Hyp Ref Expression
1 cbvralvw.1 
 |-  ( x = y -> ( ph <-> ps ) )
2 eleq1w 
 |-  ( x = y -> ( x e. A <-> y e. A ) )
3 2 1 anbi12d 
 |-  ( x = y -> ( ( x e. A /\ ph ) <-> ( y e. A /\ ps ) ) )
4 3 cbvexvw 
 |-  ( E. x ( x e. A /\ ph ) <-> E. y ( y e. A /\ ps ) )
5 df-rex 
 |-  ( E. x e. A ph <-> E. x ( x e. A /\ ph ) )
6 df-rex 
 |-  ( E. y e. A ps <-> E. y ( y e. A /\ ps ) )
7 4 5 6 3bitr4i 
 |-  ( E. x e. A ph <-> E. y e. A ps )