Metamath Proof Explorer


Theorem r19.28z

Description: Restricted quantifier version of Theorem 19.28 of Margaris p. 90. It is valid only when the domain of quantification is not empty. (Contributed by NM, 26-Oct-2010)

Ref Expression
Hypothesis r19.3rz.1
|- F/ x ph
Assertion r19.28z
|- ( A =/= (/) -> ( A. x e. A ( ph /\ ps ) <-> ( ph /\ A. x e. A ps ) ) )

Proof

Step Hyp Ref Expression
1 r19.3rz.1
 |-  F/ x ph
2 r19.26
 |-  ( A. x e. A ( ph /\ ps ) <-> ( A. x e. A ph /\ A. x e. A ps ) )
3 1 r19.3rz
 |-  ( A =/= (/) -> ( ph <-> A. x e. A ph ) )
4 3 anbi1d
 |-  ( A =/= (/) -> ( ( ph /\ A. x e. A ps ) <-> ( A. x e. A ph /\ A. x e. A ps ) ) )
5 2 4 bitr4id
 |-  ( A =/= (/) -> ( A. x e. A ( ph /\ ps ) <-> ( ph /\ A. x e. A ps ) ) )