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Theorem 19.38 1667
Description: Theorem 19.38 of [Margaris] p. 90. (Contributed by NM, 12-Mar-1993.) Allow a shortening of 19.21t 1909 and 19.23t 1914. (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
StepHypRef Expression
1 alnex 1619 . . 3  |-  ( A. x  -.  ph  <->  -.  E. x ph )
2 pm2.21 108 . . . 4  |-  ( -. 
ph  ->  ( ph  ->  ps ) )
32alimi 1638 . . 3  |-  ( A. x  -.  ph  ->  A. x
( ph  ->  ps )
)
41, 3sylbir 213 . 2  |-  ( -. 
E. x ph  ->  A. x ( ph  ->  ps ) )
5 ala1 1665 . 2  |-  ( A. x ps  ->  A. x
( ph  ->  ps )
)
64, 5ja 161 1  |-  ( ( E. x ph  ->  A. x ps )  ->  A. x ( ph  ->  ps ) )
Colors of variables: wff setvar class
Syntax hints:   -. wn 3    -> wi 4   A.wal 1396   E.wex 1617
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1623  ax-4 1636
This theorem depends on definitions:  df-bi 185  df-ex 1618
This theorem is referenced by:  19.21v  1734  19.23v  1765  19.21t  1909  19.23t  1914  pm10.53  31512  bj-19.21t  34804
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