Theorem 19.9t 1635

Description: A closed version of 19.9 1637. (Contributed by NM, 5-Aug-1993.) (Revised by Mario Carneiro, 24-Sep-2016.) (Proof shortended by Wolf Lammen, 30-Dec-2017.)

Assertion

Ref	Expression
19.9t	|- ( F/ x ph -> ( E. x ph <-> ph ) )

Proof of Theorem 19.9t

Step	Hyp	Ref	Expression
1		df-nf 1454	. . 3 |- ( F/ x ph <-> A. x ( ph -> A. x ph ) )
2		19.9ht 1634	. . 3 |- ( A. x ( ph -> A. x ph ) -> ( E. x ph -> ph ) )
3	1, 2	sylbi 120	. 2 |- ( F/ x ph -> ( E. x ph -> ph ) )
4		19.8a 1583	. 2 |- ( ph -> E. x ph )
5	3, 4	impbid1 141	1 |- ( F/ x ph -> ( E. x ph <-> ph ) )

Syntax hints:   -> wi 4   <-> wb 104   A.wal 1346   F/wnf 1453   E.wex 1485

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-gen 1442  ax-ie1 1486  ax-ie2 1487  ax-4 1503

This theorem depends on definitions:  df-bi 116  df-nf 1454

This theorem is referenced by:  19.9d  1654  19.23t  1670  spimt  1729  exdistrfor  1793  sbequi  1832  sbft  1841  vtoclegft  2802  copsexg  4229