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Theorem 19.23t 1612
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 1535 . . 3  |-  ( A. x ( ph  ->  ps )  ->  ( E. x ph  ->  E. x ps ) )
2 19.9t 1578 . . . 4  |-  ( F/ x ps  ->  ( E. x ps  <->  ps )
)
32biimpd 142 . . 3  |-  ( F/ x ps  ->  ( E. x ps  ->  ps ) )
41, 3syl9r 72 . 2  |-  ( F/ x ps  ->  ( A. x ( ph  ->  ps )  ->  ( E. x ph  ->  ps )
) )
5 nfr 1456 . . . 4  |-  ( F/ x ps  ->  ( ps  ->  A. x ps )
)
65imim2d 53 . . 3  |-  ( F/ x ps  ->  (
( E. x ph  ->  ps )  ->  ( E. x ph  ->  A. x ps ) ) )
7 19.38 1611 . . 3  |-  ( ( E. x ph  ->  A. x ps )  ->  A. x ( ph  ->  ps ) )
86, 7syl6 33 . 2  |-  ( F/ x ps  ->  (
( E. x ph  ->  ps )  ->  A. x
( ph  ->  ps )
) )
94, 8impbid 127 1  |-  ( F/ x ps  ->  ( A. x ( ph  ->  ps )  <->  ( E. x ph  ->  ps ) ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 103   A.wal 1287   F/wnf 1394   E.wex 1426
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106  ax-5 1381  ax-gen 1383  ax-ie1 1427  ax-ie2 1428  ax-4 1445  ax-ial 1472  ax-i5r 1473
This theorem depends on definitions:  df-bi 115  df-nf 1395
This theorem is referenced by:  19.23  1613  r19.23t  2479  ceqsalt  2645  vtoclgft  2669  sbciegft  2867
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