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Theorem bj-hbntbi 34659
Description: Strengthening hbnt 1899 by replacing its succedent with a biconditional. See also hbntg 29478 and hbntal 33720. (Contributed by BJ, 20-Oct-2019.) Proved from bj-19.9htbi 34658. (Proof modification is discouraged.)
Assertion
Ref Expression
bj-hbntbi  |-  ( A. x ( ph  ->  A. x ph )  -> 
( -.  ph  <->  A. x  -.  ph ) )

Proof of Theorem bj-hbntbi
StepHypRef Expression
1 bj-19.9htbi 34658 . . . 4  |-  ( A. x ( ph  ->  A. x ph )  -> 
( E. x ph  <->  ph ) )
21bicomd 201 . . 3  |-  ( A. x ( ph  ->  A. x ph )  -> 
( ph  <->  E. x ph )
)
32notbid 292 . 2  |-  ( A. x ( ph  ->  A. x ph )  -> 
( -.  ph  <->  -.  E. x ph ) )
4 alnex 1619 . 2  |-  ( A. x  -.  ph  <->  -.  E. x ph )
53, 4syl6bbr 263 1  |-  ( A. x ( ph  ->  A. x ph )  -> 
( -.  ph  <->  A. x  -.  ph ) )
Colors of variables: wff setvar class
Syntax hints:   -. wn 3    -> wi 4    <-> wb 184   A.wal 1396   E.wex 1617
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1623  ax-4 1636  ax-5 1709  ax-6 1752  ax-7 1795  ax-10 1842  ax-12 1859
This theorem depends on definitions:  df-bi 185  df-ex 1618
This theorem is referenced by: (None)
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