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Theorem ax6blem 1522
Description: If x is not free in φ, it is not free in ¬ φ. This theorem doesn't use ax6b 1523 compared to hbnt 1525. (Contributed by GD, 27-Jan-2018.)
Hypothesis
Ref Expression
ax6blem.1 (φxφ)
Assertion
Ref Expression
ax6blem φx ¬ φ)

Proof of Theorem ax6blem
StepHypRef Expression
1 ax6blem.1 . . . 4 (φxφ)
2 id 19 . . . 4 (φφ)
31, 2exlimih 1466 . . 3 (xφφ)
43con3i 549 . 2 φ → ¬ xφ)
5 alnex 1369 . 2 (x ¬ φ ↔ ¬ xφ)
64, 5sylibr 137 1 φx ¬ φ)
Colors of variables: wff set class
Syntax hints:  ¬ wn 3  wi 4  wal 1226  wex 1362
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 99  ax-ia2 100  ax-ia3 101  ax-in1 532  ax-in2 533  ax-5 1316  ax-gen 1318  ax-ie2 1364
This theorem depends on definitions:  df-bi 110  df-tru 1231  df-fal 1234
This theorem is referenced by:  ax6b  1523
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