Theorem axie2 2597

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

Assertion

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

Proof of Theorem axie2

Step	Hyp	Ref	Expression
1		nf5 2116	. 2 |- ( F/ x ps <-> A. x ( ps -> A. x ps ) )
2		19.23t 2079	. 2 |- ( F/ x ps -> ( A. x ( ph -> ps ) <-> ( E. x ph -> ps ) ) )
3	1, 2	sylbir 225	1 |- ( A. x ( ps -> A. x ps ) -> ( A. x ( ph -> ps ) <-> ( E. x ph -> ps ) ) )

Syntax hints:   -> wi 4   <-> wb 196   A.wal 1481   E.wex 1704   F/wnf 1708

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1722  ax-4 1737  ax-5 1839  ax-6 1888  ax-7 1935  ax-10 2019  ax-12 2047

This theorem depends on definitions:  df-bi 197  df-or 385  df-ex 1705  df-nf 1710

This theorem is referenced by: (None)