Metamath Proof Explorer


Theorem 19.38

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

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

Proof

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