Metamath Proof Explorer

Theorem r19.9rzv

Description: Restricted quantification of wff not containing quantified variable. (Contributed by NM, 27-May-1998)

Ref Expression
Assertion r19.9rzv 
|- ( A =/= (/) -> ( ph <-> E. x e. A ph ) )

Proof

Step Hyp Ref Expression
1 dfrex2 
 |-  ( E. x e. A ph <-> -. A. x e. A -. ph )
2 r19.3rzv 
 |-  ( A =/= (/) -> ( -. ph <-> A. x e. A -. ph ) )
3 2 con1bid 
 |-  ( A =/= (/) -> ( -. A. x e. A -. ph <-> ph ) )
4 1 3 syl5rbb 
 |-  ( A =/= (/) -> ( ph <-> E. x e. A ph ) )