Theorem 19.38 1766

Description: Theorem 19.38 of [Margaris] p. 90. The converse holds under non-freeness conditions, see 19.38a 1767 and 19.38b 1768. (Contributed by NM, 12-Mar-1993.) Allow a shortening of 19.21t 2073. (Revised by Wolf Lammen, 2-Jan-2018.)

Assertion

Ref	Expression
19.38	|- ( ( E. x ph -> A. x ps ) -> A. x ( ph -> ps ) )

Proof of Theorem 19.38

Step	Hyp	Ref	Expression
1		alnex 1706	. . 3 |- ( A. x -. ph <-> -. E. x ph )
2		pm2.21 120	. . . 4 |- ( -. ph -> ( ph -> ps ) )
3	2	alimi 1739	. . 3 |- ( A. x -. ph -> A. x ( ph -> ps ) )
4	1, 3	sylbir 225	. 2 |- ( -. E. x ph -> A. x ( ph -> ps ) )
5		ala1 1741	. 2 |- ( A. x ps -> A. x ( ph -> ps ) )
6	4, 5	ja 173	1 |- ( ( E. x ph -> A. x ps ) -> A. x ( ph -> ps ) )

Syntax hints:   -. wn 3   -> wi 4   A.wal 1481   E.wex 1704

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1722  ax-4 1737

This theorem depends on definitions:  df-bi 197  df-ex 1705

This theorem is referenced by:  19.38a  1767  19.38b  1768  nfimt  1821  19.21v  1868  19.23v  1902  19.21tOLDOLD  2074  19.21tOLD  2213  bj-19.21t  32817  pm10.53  38565