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Theorem 19.29 1583
Description: Theorem 19.29 of [Margaris] p. 90. (Contributed by NM, 5-Aug-1993.) (Proof shortened by Andrew Salmon, 13-May-2011.)
Assertion
Ref Expression
19.29  |-  ( ( A. x ph  /\  E. x ps )  ->  E. x ( ph  /\  ps ) )

Proof of Theorem 19.29
StepHypRef Expression
1 pm3.2 434 . . . 4  |-  ( ph  ->  ( ps  ->  ( ph  /\  ps ) ) )
21alimi 1546 . . 3  |-  ( A. x ph  ->  A. x
( ps  ->  ( ph  /\  ps ) ) )
3 exim 1562 . . 3  |-  ( A. x ( ps  ->  (
ph  /\  ps )
)  ->  ( E. x ps  ->  E. x
( ph  /\  ps )
) )
42, 3syl 15 . 2  |-  ( A. x ph  ->  ( E. x ps  ->  E. x
( ph  /\  ps )
) )
54imp 418 1  |-  ( ( A. x ph  /\  E. x ps )  ->  E. x ( ph  /\  ps ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 358   A.wal 1527   E.wex 1528
This theorem is referenced by:  19.29r  1584  19.29x  1586  equs4  1899  equvini  1927  supsrlem  8733  1stccnp  17188  iscmet3  18719  isch3  21821  stoweidlem35  27784  bnj849  28957
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1533  ax-5 1544
This theorem depends on definitions:  df-bi 177  df-an 360  df-ex 1529
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