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Theorem 19.21t 1809
Description: Closed form of Theorem 19.21 of [Margaris] p. 90. (Contributed by NM, 27-May-1997.) (Revised by Mario Carneiro, 24-Sep-2016.) (Proof shortened by Wolf Lammen, 3-Jan-2018.)
Assertion
Ref Expression
19.21t  |-  ( F/ x ph  ->  ( A. x ( ph  ->  ps )  <->  ( ph  ->  A. x ps ) ) )

Proof of Theorem 19.21t
StepHypRef Expression
1 nfr 1773 . . 3  |-  ( F/ x ph  ->  ( ph  ->  A. x ph )
)
2 ax-5 1563 . . 3  |-  ( A. x ( ph  ->  ps )  ->  ( A. x ph  ->  A. x ps ) )
31, 2syl9 68 . 2  |-  ( F/ x ph  ->  ( A. x ( ph  ->  ps )  ->  ( ph  ->  A. x ps )
) )
4 19.9t 1789 . . . 4  |-  ( F/ x ph  ->  ( E. x ph  <->  ph ) )
54imbi1d 309 . . 3  |-  ( F/ x ph  ->  (
( E. x ph  ->  A. x ps )  <->  (
ph  ->  A. x ps )
) )
6 19.38 1808 . . 3  |-  ( ( E. x ph  ->  A. x ps )  ->  A. x ( ph  ->  ps ) )
75, 6syl6bir 221 . 2  |-  ( F/ x ph  ->  (
( ph  ->  A. x ps )  ->  A. x
( ph  ->  ps )
) )
83, 7impbid 184 1  |-  ( F/ x ph  ->  ( A. x ( ph  ->  ps )  <->  ( ph  ->  A. x ps ) ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 177   A.wal 1546   E.wex 1547   F/wnf 1550
This theorem is referenced by:  19.21  1810  nfimd  1823  sbcom  2138  sbal2  2184  ax11indalem  2247  ax11inda2ALT  2248  r19.21t  2751  ceqsalt  2938  sbciegft  3151  sbcomwAUX7  29291  sbcomOLD7  29439  sbal2OLD7  29453
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1552  ax-5 1563  ax-17 1623  ax-9 1662  ax-8 1683  ax-6 1740  ax-11 1757
This theorem depends on definitions:  df-bi 178  df-ex 1548  df-nf 1551
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