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Theorem hbnt 1800
Description: Closed theorem version of bound-variable hypothesis builder hbn 1802. (Contributed by NM, 5-Aug-1993.) (Proof shortened by Wolf Lammen, 3-Mar-2018.)
Assertion
Ref Expression
hbnt  |-  ( A. x ( ph  ->  A. x ph )  -> 
( -.  ph  ->  A. x  -.  ph )
)

Proof of Theorem hbnt
StepHypRef Expression
1 df-ex 1552 . . 3  |-  ( E. x ph  <->  -.  A. x  -.  ph )
2 19.9ht 1793 . . 3  |-  ( A. x ( ph  ->  A. x ph )  -> 
( E. x ph  ->  ph ) )
31, 2syl5bir 211 . 2  |-  ( A. x ( ph  ->  A. x ph )  -> 
( -.  A. x  -.  ph  ->  ph ) )
43con1d 119 1  |-  ( A. x ( ph  ->  A. x ph )  -> 
( -.  ph  ->  A. x  -.  ph )
)
Colors of variables: wff set class
Syntax hints:   -. wn 3    -> wi 4   A.wal 1550   E.wex 1551
This theorem is referenced by:  hbn  1802  19.9htOLD  1804  nfnd  1810  nfimdOLD  1829  hbnd  1906
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1556  ax-5 1567  ax-17 1627  ax-9 1667  ax-8 1688  ax-6 1745  ax-11 1762
This theorem depends on definitions:  df-bi 179  df-ex 1552
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