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Metamath Proof Explorer

Theorem r19.36vf

Description: Restricted quantifier version of one direction of 19.36 . (Contributed by Glauco Siliprandi, 23-Oct-2021)

Ref Expression
Hypothesis r19.36vf.1 
|- F/ x ps
Assertion r19.36vf 
|- ( E. x e. A ( ph -> ps ) -> ( A. x e. A ph -> ps ) )

Proof

Step Hyp Ref Expression
1 r19.36vf.1 
 |-  F/ x ps
2 r19.35 
 |-  ( E. x e. A ( ph -> ps ) <-> ( A. x e. A ph -> E. x e. A ps ) )
3 idd 
 |-  ( x e. A -> ( ps -> ps ) )
4 1 3 rexlimi 
 |-  ( E. x e. A ps -> ps )
5 4 imim2i 
 |-  ( ( A. x e. A ph -> E. x e. A ps ) -> ( A. x e. A ph -> ps ) )
6 2 5 sylbi 
 |-  ( E. x e. A ( ph -> ps ) -> ( A. x e. A ph -> ps ) )