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Theorem 19.21t 1770
Description: Closed form of Theorem 19.21 of [Margaris] p. 90. (Contributed by NM, 27-May-1997.) (Revised by Mario Carneiro, 24-Sep-2016.)
Assertion
Ref Expression
19.21t  |-  ( F/ x ph  ->  ( A. x ( ph  ->  ps )  <->  ( ph  ->  A. x ps ) ) )

Proof of Theorem 19.21t
StepHypRef Expression
1 id 21 . . . 4  |-  ( F/ x ph  ->  F/ x ph )
21nfrd 1704 . . 3  |-  ( F/ x ph  ->  ( ph  ->  A. x ph )
)
3 alim 1548 . . 3  |-  ( A. x ( ph  ->  ps )  ->  ( A. x ph  ->  A. x ps ) )
42, 3syl9 68 . 2  |-  ( F/ x ph  ->  ( A. x ( ph  ->  ps )  ->  ( ph  ->  A. x ps )
) )
5 nfa1 1719 . . . . . 6  |-  F/ x A. x ps
65a1i 12 . . . . 5  |-  ( F/ x ph  ->  F/ x A. x ps )
71, 6nfimd 1727 . . . 4  |-  ( F/ x ph  ->  F/ x ( ph  ->  A. x ps ) )
87nfrd 1704 . . 3  |-  ( F/ x ph  ->  (
( ph  ->  A. x ps )  ->  A. x
( ph  ->  A. x ps ) ) )
9 ax-4 1692 . . . . 5  |-  ( A. x ps  ->  ps )
109imim2i 15 . . . 4  |-  ( (
ph  ->  A. x ps )  ->  ( ph  ->  ps ) )
1110alimi 1546 . . 3  |-  ( A. x ( ph  ->  A. x ps )  ->  A. x ( ph  ->  ps ) )
128, 11syl6 31 . 2  |-  ( F/ x ph  ->  (
( ph  ->  A. x ps )  ->  A. x
( ph  ->  ps )
) )
134, 12impbid 185 1  |-  ( F/ x ph  ->  ( A. x ( ph  ->  ps )  <->  ( ph  ->  A. x ps ) ) )
Colors of variables: wff set class
Syntax hints:    -> wi 6    <-> wb 178   A.wal 1532   F/wnf 1539
This theorem is referenced by:  19.21  1771  sbcom  1984  sbal2  2103  ax11indalem  2113  ax11inda2ALT  2114  r19.21t  2599  ceqsalt  2761  sbciegft  2965
This theorem was proved from axioms:  ax-1 7  ax-2 8  ax-3 9  ax-mp 10  ax-5 1533  ax-6 1534  ax-gen 1536  ax-4 1692
This theorem depends on definitions:  df-bi 179  df-an 362  df-nf 1540
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