Metamath Proof Explorer


Theorem r19.37zv

Description: Restricted quantifier version of Theorem 19.37 of Margaris p. 90. It is valid only when the domain of quantification is not empty. (Contributed by Paul Chapman, 8-Oct-2007)

Ref Expression
Assertion r19.37zv
|- ( A =/= (/) -> ( E. x e. A ( ph -> ps ) <-> ( ph -> E. x e. A ps ) ) )

Proof

Step Hyp Ref Expression
1 r19.35
 |-  ( E. x e. A ( ph -> ps ) <-> ( A. x e. A ph -> E. x e. A ps ) )
2 r19.3rzv
 |-  ( A =/= (/) -> ( ph <-> A. x e. A ph ) )
3 2 imbi1d
 |-  ( A =/= (/) -> ( ( ph -> E. x e. A ps ) <-> ( A. x e. A ph -> E. x e. A ps ) ) )
4 1 3 bitr4id
 |-  ( A =/= (/) -> ( E. x e. A ( ph -> ps ) <-> ( ph -> E. x e. A ps ) ) )