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Theorem hbntal 28640
Description: A closed form of hbn 1801. hbnt 1799 is another closed form of hbn 1801. (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 1804 . 2  |-  ( A. x ( ph  ->  A. x ph )  ->  A. x A. x (
ph  ->  A. x ph )
)
2 ax6o 1766 . . . . 5  |-  ( -. 
A. x  -.  A. x ph  ->  ph )
32con1i 123 . . . 4  |-  ( -. 
ph  ->  A. x  -.  A. x ph )
4 con3 128 . . . . 5  |-  ( (
ph  ->  A. x ph )  ->  ( -.  A. x ph  ->  -.  ph ) )
54al2imi 1570 . . . 4  |-  ( A. x ( ph  ->  A. x ph )  -> 
( A. x  -.  A. x ph  ->  A. x  -.  ph ) )
63, 5syl5 30 . . 3  |-  ( A. x ( ph  ->  A. x ph )  -> 
( -.  ph  ->  A. x  -.  ph )
)
76alimi 1568 . 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 set class
Syntax hints:   -. wn 3    -> wi 4   A.wal 1549
This theorem is referenced by:  hbimpg  28641  hbimpgVD  29016  hbexgVD  29018
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1555  ax-5 1566  ax-17 1626  ax-9 1666  ax-8 1687  ax-6 1744  ax-11 1761
This theorem depends on definitions:  df-bi 178  df-ex 1551
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