Theorem 19.9ht 1588

Description: A closed version of one direction of 19.9 1591. (Contributed by NM, 5-Aug-1993.)

Assertion

Ref	Expression
19.9ht	|- ( A. x ( ph -> A. x ph ) -> ( E. x ph -> ph ) )

Proof of Theorem 19.9ht

Step	Hyp	Ref	Expression
1		id 19	. . 3 |- ( ph -> ph )
2	1	ax-gen 1393	. 2 |- A. x ( ph -> ph )
3		19.23ht 1441	. 2 |- ( A. x ( ph -> A. x ph ) -> ( A. x ( ph -> ph ) <-> ( E. x ph -> ph ) ) )
4	2, 3	mpbii 147	1 |- ( A. x ( ph -> A. x ph ) -> ( E. x ph -> ph ) )

Syntax hints:   -> wi 4   A.wal 1297   E.wex 1436

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 105  ax-gen 1393  ax-ie2 1438

This theorem depends on definitions:  df-bi 116

This theorem is referenced by:  19.9t  1589  19.9h  1590  19.9hd  1608