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Theorem 19.21t 1802
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 19 . . . 4  |-  ( F/ x ph  ->  F/ x ph )
21nfrd 1755 . . 3  |-  ( F/ x ph  ->  ( ph  ->  A. x ph )
)
3 alim 1548 . . 3  |-  ( A. x ( ph  ->  ps )  ->  ( A. x ph  ->  A. x ps ) )
42, 3syl9 66 . 2  |-  ( F/ x ph  ->  ( A. x ( ph  ->  ps )  ->  ( ph  ->  A. x ps )
) )
5 nfa1 1768 . . . . . 6  |-  F/ x A. x ps
65a1i 10 . . . . 5  |-  ( F/ x ph  ->  F/ x A. x ps )
71, 6nfimd 1773 . . . 4  |-  ( F/ x ph  ->  F/ x ( ph  ->  A. x ps ) )
87nfrd 1755 . . 3  |-  ( F/ x ph  ->  (
( ph  ->  A. x ps )  ->  A. x
( ph  ->  A. x ps ) ) )
9 sp 1728 . . . . 5  |-  ( A. x ps  ->  ps )
109imim2i 13 . . . 4  |-  ( (
ph  ->  A. x ps )  ->  ( ph  ->  ps ) )
1110alimi 1549 . . 3  |-  ( A. x ( ph  ->  A. x ps )  ->  A. x ( ph  ->  ps ) )
128, 11syl6 29 . 2  |-  ( F/ x ph  ->  (
( ph  ->  A. x ps )  ->  A. x
( ph  ->  ps )
) )
134, 12impbid 183 1  |-  ( F/ x ph  ->  ( A. x ( ph  ->  ps )  <->  ( ph  ->  A. x ps ) ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 176   A.wal 1530   F/wnf 1534
This theorem is referenced by:  19.21  1803  sbcom  2042  sbal2  2086  ax11indalem  2149  ax11inda2ALT  2150  r19.21t  2641  ceqsalt  2823  sbciegft  3034  sbcomwAUX7  29562  sbcomOLD7  29709  sbal2OLD7  29724
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1536  ax-5 1547  ax-17 1606  ax-9 1644  ax-8 1661  ax-6 1715  ax-11 1727
This theorem depends on definitions:  df-bi 177  df-an 360  df-nf 1535
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