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Theorem hbnt 1653
Description: Closed theorem version of bound-variable hypothesis builder hbn 1654. (Contributed by NM, 5-Aug-1993.) (Revised by NM, 2-Feb-2015.)
Assertion
Ref Expression
hbnt  |-  ( A. x ( ph  ->  A. x ph )  -> 
( -.  ph  ->  A. x  -.  ph )
)

Proof of Theorem hbnt
StepHypRef Expression
1 ax-4 1510 . . . 4  |-  ( A. x ph  ->  ph )
21con3i 632 . . 3  |-  ( -. 
ph  ->  -.  A. x ph )
3 ax6b 1651 . . 3  |-  ( -. 
A. x ph  ->  A. x  -.  A. x ph )
42, 3syl 14 . 2  |-  ( -. 
ph  ->  A. x  -.  A. x ph )
5 con3 642 . . 3  |-  ( (
ph  ->  A. x ph )  ->  ( -.  A. x ph  ->  -.  ph ) )
65al2imi 1458 . 2  |-  ( A. x ( ph  ->  A. x ph )  -> 
( A. x  -.  A. x ph  ->  A. x  -.  ph ) )
74, 6syl5 32 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 1351
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-in1 614  ax-in2 615  ax-5 1447  ax-gen 1449  ax-ie2 1494  ax-4 1510  ax-ial 1534
This theorem depends on definitions:  df-bi 117  df-tru 1356  df-fal 1359
This theorem is referenced by:  hbn  1654  hbnd  1655  nfnt  1656
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