MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  hbnt Unicode version

Theorem hbnt 1736
Description: Closed theorem version of bound-variable hypothesis builder hbn 1732. (Contributed by NM, 5-Aug-1993.)
Assertion
Ref Expression
hbnt  |-  ( A. x ( ph  ->  A. x ph )  -> 
( -.  ph  ->  A. x  -.  ph )
)

Proof of Theorem hbnt
StepHypRef Expression
1 ax6o 1735 . . 3  |-  ( -. 
A. x  -.  A. x ph  ->  ph )
21con1i 121 . 2  |-  ( -. 
ph  ->  A. x  -.  A. x ph )
3 con3 126 . . 3  |-  ( (
ph  ->  A. x ph )  ->  ( -.  A. x ph  ->  -.  ph ) )
43al2imi 1551 . 2  |-  ( A. x ( ph  ->  A. x ph )  -> 
( A. x  -.  A. x ph  ->  A. x  -.  ph ) )
52, 4syl5 28 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 1530
This theorem is referenced by:  19.9ht  1738  nfimd  1773  hbnd  1836
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1536  ax-5 1547  ax-17 1606  ax-9 1644  ax-8 1661  ax-6 1715  ax-11 1727
  Copyright terms: Public domain W3C validator