Theorem i19.39 1654

Description: Theorem 19.39 of [Margaris] p. 90, with an additional hypothesis. The hypothesis is the converse of 19.35-1 1638, and is a theorem of classical logic, but in intuitionistic logic it will only be provable for some propositions. (Contributed by Jim Kingdon, 22-Jul-2018.)

Hypothesis

Ref	Expression
i19.24.1	|- ( ( A. x ph -> E. x ps ) -> E. x ( ph -> ps ) )

Assertion

Ref	Expression
i19.39	|- ( ( E. x ph -> E. x ps ) -> E. x ( ph -> ps ) )

Proof of Theorem i19.39

Step	Hyp	Ref	Expression
1		19.2 1652	. . 3 |- ( A. x ph -> E. x ph )
2	1	imim1i 60	. 2 |- ( ( E. x ph -> E. x ps ) -> ( A. x ph -> E. x ps ) )
3		i19.24.1	. 2 |- ( ( A. x ph -> E. x ps ) -> E. x ( ph -> ps ) )
4	2, 3	syl 14	1 |- ( ( E. x ph -> E. x ps ) -> E. x ( ph -> ps ) )

Syntax hints:   -> wi 4   A.wal 1362   E.wex 1506

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-gen 1463  ax-ie1 1507  ax-ie2 1508  ax-4 1524

This theorem depends on definitions:  df-bi 117

This theorem is referenced by: (None)