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Theorem hbimtg 29482
Description: A more general and closed form of hbim 1927. (Contributed by Scott Fenton, 13-Dec-2010.)
Assertion
Ref Expression
hbimtg  |-  ( ( A. x ( ph  ->  A. x ch )  /\  ( ps  ->  A. x th ) )  ->  (
( ch  ->  ps )  ->  A. x ( ph  ->  th ) ) )

Proof of Theorem hbimtg
StepHypRef Expression
1 hbntg 29481 . . . 4  |-  ( A. x ( ph  ->  A. x ch )  -> 
( -.  ch  ->  A. x  -.  ph )
)
2 pm2.21 108 . . . . 5  |-  ( -. 
ph  ->  ( ph  ->  th ) )
32alimi 1638 . . . 4  |-  ( A. x  -.  ph  ->  A. x
( ph  ->  th )
)
41, 3syl6 33 . . 3  |-  ( A. x ( ph  ->  A. x ch )  -> 
( -.  ch  ->  A. x ( ph  ->  th ) ) )
54adantr 463 . 2  |-  ( ( A. x ( ph  ->  A. x ch )  /\  ( ps  ->  A. x th ) )  ->  ( -.  ch  ->  A. x
( ph  ->  th )
) )
6 ala1 1665 . . . 4  |-  ( A. x th  ->  A. x
( ph  ->  th )
)
76imim2i 14 . . 3  |-  ( ( ps  ->  A. x th )  ->  ( ps 
->  A. x ( ph  ->  th ) ) )
87adantl 464 . 2  |-  ( ( A. x ( ph  ->  A. x ch )  /\  ( ps  ->  A. x th ) )  ->  ( ps  ->  A. x ( ph  ->  th ) ) )
95, 8jad 162 1  |-  ( ( A. x ( ph  ->  A. x ch )  /\  ( ps  ->  A. x th ) )  ->  (
( ch  ->  ps )  ->  A. x ( ph  ->  th ) ) )
Colors of variables: wff setvar class
Syntax hints:   -. wn 3    -> wi 4    /\ wa 367   A.wal 1396
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-an 369  df-ex 1618
This theorem is referenced by:  hbimg  29485
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