Theorem 19.9t 1533

Description: A closed version of 19.9 1535. (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 1350	. . 3 |- ( F/ x ph <-> A. x ( ph -> A. x ph ) )
2		19.9ht 1532	. . 3 |- ( A. x ( ph -> A. x ph ) -> ( E. x ph -> ph ) )
3	1, 2	sylbi 114	. 2 |- ( F/ x ph -> ( E. x ph -> ph ) )
4		19.8a 1482	. 2 |- ( ph -> E. x ph )
5	3, 4	impbid1 130	1 |- ( F/ x ph -> ( E. x ph <-> ph ) )

Syntax hints:   -> wi 4   <-> wb 98   A.wal 1241   F/wnf 1349   E.wex 1381

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 99  ax-ia2 100  ax-ia3 101  ax-gen 1338  ax-ie1 1382  ax-ie2 1383  ax-4 1400

This theorem depends on definitions:  df-bi 110  df-nf 1350

This theorem is referenced by:  19.9d  1551  19.23t  1567  spimt  1624  exdistrfor  1681  sbequi  1720  sbft  1728  vtoclegft  2625  copsexg  3981