Theorem 19.21t 1843

Description: Closed form of Theorem 19.21 of [Margaris] p. 90. (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 1812	. . 3 |- ( F/ x ph -> ( ph -> A. x ph ) )
2		alim 1604	. . 3 |- ( A. x ( ph -> ps ) -> ( A. x ph -> A. x ps ) )
3	1, 2	syl9 71	. 2 |- ( F/ x ph -> ( A. x ( ph -> ps ) -> ( ph -> A. x ps ) ) )
4		19.9t 1829	. . . 4 |- ( F/ x ph -> ( E. x ph <-> ph ) )
5	4	imbi1d 317	. . 3 |- ( F/ x ph -> ( ( E. x ph -> A. x ps ) <-> ( ph -> A. x ps ) ) )
6		19.38 1630	. . 3 |- ( ( E. x ph -> A. x ps ) -> A. x ( ph -> ps ) )
7	5, 6	syl6bir 229	. 2 |- ( F/ x ph -> ( ( ph -> A. x ps ) -> A. x ( ph -> ps ) ) )
8	3, 7	impbid 191	1 |- ( F/ x ph -> ( A. x ( ph -> ps ) <-> ( ph -> A. x ps ) ) )

Syntax hints:   -> wi 4   <-> wb 184   A.wal 1368   E.wex 1587   F/wnf 1590

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1592  ax-4 1603  ax-5 1671  ax-6 1710  ax-7 1730  ax-10 1777  ax-12 1794

This theorem depends on definitions:  df-bi 185  df-ex 1588  df-nf 1591

This theorem is referenced by:  19.21  1844  nfimd  1855  sbal1  2180  sbal2  2182  sbal2OLD  2193  ax12indalem  2255  ax12inda2ALT  2256  r19.21t  2907  ceqsalt  3101  sbciegft  3325  wl-sbhbt  28527  wl-2sb6d  28533  wl-sbalnae  28537  bj-ceqsalt0  32717  bj-ceqsalt1  32718