Metamath Proof Explorer

Theorem sbcrex

Description: Interchange class substitution and restricted existential quantifier. (Contributed by NM, 15-Nov-2005) (Revised by NM, 18-Aug-2018)

Ref Expression
Assertion sbcrex 
|- ( [. A / x ]. E. y e. B ph <-> E. y e. B [. A / x ]. ph )

Proof

Step Hyp Ref Expression
1 nfcv 
 |-  F/_ y A
2 sbcrext 
 |-  ( F/_ y A -> ( [. A / x ]. E. y e. B ph <-> E. y e. B [. A / x ]. ph ) )
3 1 2 ax-mp 
 |-  ( [. A / x ]. E. y e. B ph <-> E. y e. B [. A / x ]. ph )