Metamath Proof Explorer

Theorem r19.29r

Description: Restricted quantifier version of 19.29r ; variation of r19.29 . (Contributed by NM, 31-Aug-1999) (Proof shortened by Wolf Lammen, 29-Jun-2023)

Ref Expression
Assertion r19.29r 
|- ( ( E. x e. A ph /\ A. x e. A ps ) -> E. x e. A ( ph /\ ps ) )

Proof

Step Hyp Ref Expression
1 iba 
 |-  ( ps -> ( ph <-> ( ph /\ ps ) ) )
2 1 ralrexbid 
 |-  ( A. x e. A ps -> ( E. x e. A ph <-> E. x e. A ( ph /\ ps ) ) )
3 2 biimpac 
 |-  ( ( E. x e. A ph /\ A. x e. A ps ) -> E. x e. A ( ph /\ ps ) )