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Theorem 19.38 1812
Description: Theorem 19.38 of [Margaris] p. 90. (Contributed by NM, 5-Aug-1993.) (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 1552 . . 3  |-  ( A. x  -.  ph  <->  -.  E. x ph )
2 pm2.21 102 . . . 4  |-  ( -. 
ph  ->  ( ph  ->  ps ) )
32alimi 1568 . . 3  |-  ( A. x  -.  ph  ->  A. x
( ph  ->  ps )
)
41, 3sylbir 205 . 2  |-  ( -. 
E. x ph  ->  A. x ( ph  ->  ps ) )
5 ax-1 5 . . 3  |-  ( ps 
->  ( ph  ->  ps ) )
65alimi 1568 . 2  |-  ( A. x ps  ->  A. x
( ph  ->  ps )
)
74, 6ja 155 1  |-  ( ( E. x ph  ->  A. x ps )  ->  A. x ( ph  ->  ps ) )
Colors of variables: wff set class
Syntax hints:   -. wn 3    -> wi 4   A.wal 1549   E.wex 1550
This theorem is referenced by:  19.21t  1813  19.23t  1818  pm10.53  27538
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1555  ax-5 1566
This theorem depends on definitions:  df-bi 178  df-ex 1551
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