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Theorem ax6blem 1628
Description: If 𝑥 is not free in 𝜑, it is not free in ¬ 𝜑. This theorem doesn't use ax6b 1629 compared to hbnt 1631. (Contributed by GD, 27-Jan-2018.)
Hypothesis
Ref Expression
ax6blem.1 (𝜑 → ∀𝑥𝜑)
Assertion
Ref Expression
ax6blem 𝜑 → ∀𝑥 ¬ 𝜑)

Proof of Theorem ax6blem
StepHypRef Expression
1 ax6blem.1 . . . 4 (𝜑 → ∀𝑥𝜑)
2 id 19 . . . 4 (𝜑𝜑)
31, 2exlimih 1572 . . 3 (∃𝑥𝜑𝜑)
43con3i 621 . 2 𝜑 → ¬ ∃𝑥𝜑)
5 alnex 1475 . 2 (∀𝑥 ¬ 𝜑 ↔ ¬ ∃𝑥𝜑)
64, 5sylibr 133 1 𝜑 → ∀𝑥 ¬ 𝜑)
Colors of variables: wff set class
Syntax hints:  ¬ wn 3  wi 4  wal 1329  wex 1468
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-in1 603  ax-in2 604  ax-5 1423  ax-gen 1425  ax-ie2 1470
This theorem depends on definitions:  df-bi 116  df-tru 1334  df-fal 1337
This theorem is referenced by:  ax6b  1629
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