Theorem 19.21t 1809

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 1773	. . 3 |- ( F/ x ph -> ( ph -> A. x ph ) )
2		ax-5 1563	. . 3 |- ( A. x ( ph -> ps ) -> ( A. x ph -> A. x ps ) )
3	1, 2	syl9 68	. 2 |- ( F/ x ph -> ( A. x ( ph -> ps ) -> ( ph -> A. x ps ) ) )
4		19.9t 1789	. . . 4 |- ( F/ x ph -> ( E. x ph <-> ph ) )
5	4	imbi1d 309	. . 3 |- ( F/ x ph -> ( ( E. x ph -> A. x ps ) <-> ( ph -> A. x ps ) ) )
6		19.38 1808	. . 3 |- ( ( E. x ph -> A. x ps ) -> A. x ( ph -> ps ) )
7	5, 6	syl6bir 221	. 2 |- ( F/ x ph -> ( ( ph -> A. x ps ) -> A. x ( ph -> ps ) ) )
8	3, 7	impbid 184	1 |- ( F/ x ph -> ( A. x ( ph -> ps ) <-> ( ph -> A. x ps ) ) )

Syntax hints:   -> wi 4   <-> wb 177   A.wal 1546   E.wex 1547   F/wnf 1550

This theorem is referenced by:  19.21  1810  nfimd  1823  sbcom  2138  sbal2  2184  ax11indalem  2247  ax11inda2ALT  2248  r19.21t  2751  ceqsalt  2938  sbciegft  3151  sbcomwAUX7  29291  sbcomOLD7  29439  sbal2OLD7  29453

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1552  ax-5 1563  ax-17 1623  ax-9 1662  ax-8 1683  ax-6 1740  ax-11 1757

This theorem depends on definitions:  df-bi 178  df-ex 1548  df-nf 1551