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Theorem ax6blem 1540
 Description: If 𝑥 is not free in 𝜑, it is not free in ¬ 𝜑. This theorem doesn't use ax6b 1541 compared to hbnt 1543. (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 1484 . . 3 (∃𝑥𝜑𝜑)
43con3i 562 . 2 𝜑 → ¬ ∃𝑥𝜑)
5 alnex 1388 . 2 (∀𝑥 ¬ 𝜑 ↔ ¬ ∃𝑥𝜑)
64, 5sylibr 137 1 𝜑 → ∀𝑥 ¬ 𝜑)
 Colors of variables: wff set class Syntax hints:  ¬ wn 3   → wi 4  ∀wal 1241  ∃wex 1381 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 544  ax-in2 545  ax-5 1336  ax-gen 1338  ax-ie2 1383 This theorem depends on definitions:  df-bi 110  df-tru 1246  df-fal 1249 This theorem is referenced by:  ax6b  1541
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