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Theorem hbntal 36778
Description: A closed form of hbn 1950. hbnt 1949 is another closed form of hbn 1950. (Contributed by Alan Sare, 8-Feb-2014.) (Proof modification is discouraged.) (New usage is discouraged.)
Assertion
Ref Expression
hbntal  |-  ( A. x ( ph  ->  A. x ph )  ->  A. x ( -.  ph  ->  A. x  -.  ph ) )

Proof of Theorem hbntal
StepHypRef Expression
1 hba1 1951 . 2  |-  ( A. x ( ph  ->  A. x ph )  ->  A. x A. x (
ph  ->  A. x ph )
)
2 axc7 1912 . . . . 5  |-  ( -. 
A. x  -.  A. x ph  ->  ph )
32con1i 132 . . . 4  |-  ( -. 
ph  ->  A. x  -.  A. x ph )
4 con3 139 . . . . 5  |-  ( (
ph  ->  A. x ph )  ->  ( -.  A. x ph  ->  -.  ph ) )
54al2imi 1683 . . . 4  |-  ( A. x ( ph  ->  A. x ph )  -> 
( A. x  -.  A. x ph  ->  A. x  -.  ph ) )
63, 5syl5 33 . . 3  |-  ( A. x ( ph  ->  A. x ph )  -> 
( -.  ph  ->  A. x  -.  ph )
)
76alimi 1680 . 2  |-  ( A. x A. x ( ph  ->  A. x ph )  ->  A. x ( -. 
ph  ->  A. x  -.  ph ) )
81, 7syl 17 1  |-  ( A. x ( ph  ->  A. x ph )  ->  A. x ( -.  ph  ->  A. x  -.  ph ) )
Colors of variables: wff setvar class
Syntax hints:   -. wn 3    -> wi 4   A.wal 1435
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1665  ax-4 1678  ax-5 1748  ax-6 1794  ax-7 1839  ax-10 1887  ax-12 1905
This theorem depends on definitions:  df-bi 188  df-ex 1660
This theorem is referenced by:  hbimpg  36779  hbimpgVD  37162  hbexgVD  37164
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