Theorem 19.21t 1997

Description: Closed form of Theorem 19.21 of [Margaris] p. 90, see 19.21 1998. (Contributed by NM, 27-May-1997.) (Revised by Mario Carneiro, 24-Sep-2016.) (Proof shortened by Wolf Lammen, 3-Jan-2018.)

Assertion

Ref	Expression
19.21t	|- ( F/ x ph -> ( A. x ( ph -> ps ) <-> ( ph -> A. x ps ) ) )

Proof of Theorem 19.21t

Step	Hyp	Ref	Expression
1		nfr 1962	. . 3 |- ( F/ x ph -> ( ph -> A. x ph ) )
2		alim 1694	. . 3 |- ( A. x ( ph -> ps ) -> ( A. x ph -> A. x ps ) )
3	1, 2	syl9 73	. 2 |- ( F/ x ph -> ( A. x ( ph -> ps ) -> ( ph -> A. x ps ) ) )
4		19.9t 1980	. . . 4 |- ( F/ x ph -> ( E. x ph <-> ph ) )
5	4	imbi1d 323	. . 3 |- ( F/ x ph -> ( ( E. x ph -> A. x ps ) <-> ( ph -> A. x ps ) ) )
6		19.38 1723	. . 3 |- ( ( E. x ph -> A. x ps ) -> A. x ( ph -> ps ) )
7	5, 6	syl6bir 237	. 2 |- ( F/ x ph -> ( ( ph -> A. x ps ) -> A. x ( ph -> ps ) ) )
8	3, 7	impbid 195	1 |- ( F/ x ph -> ( A. x ( ph -> ps ) <-> ( ph -> A. x ps ) ) )

Syntax hints:   -> wi 4   <-> wb 189   A.wal 1453   E.wex 1674   F/wnf 1678

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1680  ax-4 1693  ax-5 1769  ax-6 1816  ax-7 1862  ax-10 1926  ax-12 1944

This theorem depends on definitions:  df-bi 190  df-an 377  df-ex 1675  df-nf 1679

This theorem is referenced by:  19.21  1998  19.23t  2002  nfimd  2011  sbal1  2300  sbal2  2301  r19.21t  2797  r19.21tOLD  2798  ceqsalt  3082  sbciegft  3310  bj-ceqsalt0  31528  bj-ceqsalt1  31529  wl-sbhbt  31928  wl-2sb6d  31934  wl-sbalnae  31938  ax12indalem  32562  ax12inda2ALT  32563