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Theorem ax6blem 1583
Description: If  x is not free in  ph, it is not free in  -.  ph. This theorem doesn't use ax6b 1584 compared to hbnt 1586. (Contributed by GD, 27-Jan-2018.)
Hypothesis
Ref Expression
ax6blem.1  |-  ( ph  ->  A. x ph )
Assertion
Ref Expression
ax6blem  |-  ( -. 
ph  ->  A. x  -.  ph )

Proof of Theorem ax6blem
StepHypRef Expression
1 ax6blem.1 . . . 4  |-  ( ph  ->  A. x ph )
2 id 19 . . . 4  |-  ( ph  ->  ph )
31, 2exlimih 1527 . . 3  |-  ( E. x ph  ->  ph )
43con3i 595 . 2  |-  ( -. 
ph  ->  -.  E. x ph )
5 alnex 1431 . 2  |-  ( A. x  -.  ph  <->  -.  E. x ph )
64, 5sylibr 132 1  |-  ( -. 
ph  ->  A. x  -.  ph )
Colors of variables: wff set class
Syntax hints:   -. wn 3    -> wi 4   A.wal 1285   E.wex 1424
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106  ax-in1 577  ax-in2 578  ax-5 1379  ax-gen 1381  ax-ie2 1426
This theorem depends on definitions:  df-bi 115  df-tru 1290  df-fal 1293
This theorem is referenced by:  ax6b  1584
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