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Theorem hbnae 2430
Description: All variables are effectively bound in a distinct variable specifier. Lemma L19 in [Megill] p. 446 (p. 14 of the preprint). Usage of this theorem is discouraged because it depends on ax-13 2370. Use the weaker hbnaev 2065 when possible. (Contributed by NM, 13-May-1993.) (New usage is discouraged.)
Assertion
Ref Expression
hbnae (¬ ∀𝑥 𝑥 = 𝑦 → ∀𝑧 ¬ ∀𝑥 𝑥 = 𝑦)

Proof of Theorem hbnae
StepHypRef Expression
1 hbae 2429 . 2 (∀𝑥 𝑥 = 𝑦 → ∀𝑧𝑥 𝑥 = 𝑦)
21hbn 2291 1 (¬ ∀𝑥 𝑥 = 𝑦 → ∀𝑧 ¬ ∀𝑥 𝑥 = 𝑦)
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wal 1539
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1913  ax-6 1971  ax-7 2011  ax-10 2137  ax-11 2154  ax-12 2171  ax-13 2370
This theorem depends on definitions:  df-bi 206  df-an 397  df-or 846  df-tru 1544  df-ex 1782  df-nf 1786
This theorem is referenced by:  hbnaes  2433  eujustALT  2570  bj-hbnaeb  35252  ax6e2nd  42782  ax6e2ndVD  43132  ax6e2ndeqVD  43133  ax6e2ndALT  43154  ax6e2ndeqALT  43155
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