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Theorem hbntal 32406
Description: A closed form of hbn 1843. hbnt 1842 is another closed form of hbn 1843. (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 1844 . 2  |-  ( A. x ( ph  ->  A. x ph )  ->  A. x A. x (
ph  ->  A. x ph )
)
2 axc7 1810 . . . . 5  |-  ( -. 
A. x  -.  A. x ph  ->  ph )
32con1i 129 . . . 4  |-  ( -. 
ph  ->  A. x  -.  A. x ph )
4 con3 134 . . . . 5  |-  ( (
ph  ->  A. x ph )  ->  ( -.  A. x ph  ->  -.  ph ) )
54al2imi 1616 . . . 4  |-  ( A. x ( ph  ->  A. x ph )  -> 
( A. x  -.  A. x ph  ->  A. x  -.  ph ) )
63, 5syl5 32 . . 3  |-  ( A. x ( ph  ->  A. x ph )  -> 
( -.  ph  ->  A. x  -.  ph )
)
76alimi 1614 . 2  |-  ( A. x A. x ( ph  ->  A. x ph )  ->  A. x ( -. 
ph  ->  A. x  -.  ph ) )
81, 7syl 16 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 1377
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1601  ax-4 1612  ax-5 1680  ax-6 1719  ax-7 1739  ax-10 1786  ax-12 1803
This theorem depends on definitions:  df-bi 185  df-ex 1597
This theorem is referenced by:  hbimpg  32407  hbimpgVD  32784  hbexgVD  32786
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