Metamath Proof Explorer

Theorem axie2

Description: A key property of existential quantification (intuitionistic logic axiom ax-ie2). (Contributed by Jim Kingdon, 31-Dec-2017)

Ref Expression
Assertion axie2 
|- ( A. x ( ps -> A. x ps ) -> ( A. x ( ph -> ps ) <-> ( E. x ph -> ps ) ) )

Proof

Step Hyp Ref Expression
1 nf5 
 |-  ( F/ x ps <-> A. x ( ps -> A. x ps ) )
2 19.23t 
 |-  ( F/ x ps -> ( A. x ( ph -> ps ) <-> ( E. x ph -> ps ) ) )
3 1 2 sylbir 
 |-  ( A. x ( ps -> A. x ps ) -> ( A. x ( ph -> ps ) <-> ( E. x ph -> ps ) ) )