Theorem 19.21t 1963

Description: Closed form of Theorem 19.21 of [Margaris] p. 90, see 19.21 1964. (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 1928	. . 3 |- ( F/ x ph -> ( ph -> A. x ph ) )
2		alim 1677	. . 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 1946	. . . 4 |- ( F/ x ph -> ( E. x ph <-> ph ) )
5	4	imbi1d 318	. . 3 |- ( F/ x ph -> ( ( E. x ph -> A. x ps ) <-> ( ph -> A. x ps ) ) )
6		19.38 1706	. . 3 |- ( ( E. x ph -> A. x ps ) -> A. x ( ph -> ps ) )
7	5, 6	syl6bir 232	. 2 |- ( F/ x ph -> ( ( ph -> A. x ps ) -> A. x ( ph -> ps ) ) )
8	3, 7	impbid 193	1 |- ( F/ x ph -> ( A. x ( ph -> ps ) <-> ( ph -> A. x ps ) ) )

Syntax hints:   -> wi 4   <-> wb 187   A.wal 1435   E.wex 1657   F/wnf 1661

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1663  ax-4 1676  ax-5 1752  ax-6 1798  ax-7 1843  ax-10 1891  ax-12 1909

This theorem depends on definitions:  df-bi 188  df-ex 1658  df-nf 1662

This theorem is referenced by:  19.21  1964  19.23t  1968  nfimd  1977  sbal1  2259  sbal2  2260  r19.21t  2819  r19.21tOLD  2820  ceqsalt  3104  sbciegft  3330  bj-ceqsalt0  31452  bj-ceqsalt1  31453  wl-sbhbt  31846  wl-2sb6d  31852  wl-sbalnae  31856  ax12indalem  32485  ax12inda2ALT  32486