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Theorem 19.23t 1818
Description: Closed form of Theorem 19.23 of [Margaris] p. 90. (Contributed by NM, 7-Nov-2005.) (Proof shortened by Wolf Lammen, 2-Jan-2018.)
Assertion
Ref Expression
19.23t  |-  ( F/ x ps  ->  ( A. x ( ph  ->  ps )  <->  ( E. x ph  ->  ps ) ) )

Proof of Theorem 19.23t
StepHypRef Expression
1 exim 1584 . . 3  |-  ( A. x ( ph  ->  ps )  ->  ( E. x ph  ->  E. x ps ) )
2 19.9t 1793 . . . 4  |-  ( F/ x ps  ->  ( E. x ps  <->  ps )
)
32biimpd 199 . . 3  |-  ( F/ x ps  ->  ( E. x ps  ->  ps ) )
41, 3syl9r 69 . 2  |-  ( F/ x ps  ->  ( A. x ( ph  ->  ps )  ->  ( E. x ph  ->  ps )
) )
5 nfr 1777 . . . 4  |-  ( F/ x ps  ->  ( ps  ->  A. x ps )
)
65imim2d 50 . . 3  |-  ( F/ x ps  ->  (
( E. x ph  ->  ps )  ->  ( E. x ph  ->  A. x ps ) ) )
7 19.38 1812 . . 3  |-  ( ( E. x ph  ->  A. x ps )  ->  A. x ( ph  ->  ps ) )
86, 7syl6 31 . 2  |-  ( F/ x ps  ->  (
( E. x ph  ->  ps )  ->  A. x
( ph  ->  ps )
) )
94, 8impbid 184 1  |-  ( F/ x ps  ->  ( A. x ( ph  ->  ps )  <->  ( E. x ph  ->  ps ) ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 177   A.wal 1549   E.wex 1550   F/wnf 1553
This theorem is referenced by:  19.23  1819  sbftOLD  2116  axie2  2412  r19.23t  2820  ceqsalt  2978  vtoclgft  3002  sbciegft  3191  sbftNEW7  29556
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1555  ax-5 1566  ax-17 1626  ax-9 1666  ax-8 1687  ax-6 1744  ax-11 1761
This theorem depends on definitions:  df-bi 178  df-ex 1551  df-nf 1554
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