Theorem 19.9ht 1655

Description: A closed version of one direction of 19.9 1658. (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 1463	. 2 |- A. x ( ph -> ph )
3		19.23ht 1511	. 2 |- ( A. x ( ph -> A. x ph ) -> ( A. x ( ph -> ph ) <-> ( E. x ph -> ph ) ) )
4	2, 3	mpbii 148	1 |- ( A. x ( ph -> A. x ph ) -> ( E. x ph -> ph ) )

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-gen 1463  ax-ie2 1508

This theorem depends on definitions:  df-bi 117

This theorem is referenced by:  19.9t  1656  19.9h  1657  19.9hd  1676