Theorem 19.9t 1688

Description: A closed version of 19.9 1690. (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 1507	. . 3 |- ( F/ x ph <-> A. x ( ph -> A. x ph ) )
2		19.9ht 1687	. . 3 |- ( A. x ( ph -> A. x ph ) -> ( E. x ph -> ph ) )
3	1, 2	sylbi 121	. 2 |- ( F/ x ph -> ( E. x ph -> ph ) )
4		19.8a 1636	. 2 |- ( ph -> E. x ph )
5	3, 4	impbid1 142	1 |- ( F/ x ph -> ( E. x ph <-> ph ) )

Syntax hints:   -> wi 4   <-> wb 105   A.wal 1393   F/wnf 1506   E.wex 1538

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 1495  ax-ie1 1539  ax-ie2 1540  ax-4 1556

This theorem depends on definitions:  df-bi 117  df-nf 1507

This theorem is referenced by:  19.9d  1707  19.23t  1723  spimt  1782  exdistrfor  1846  sbequi  1885  sbft  1894  vtoclegft  2875  copsexg  4330