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Theorem hbnae 2423
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 2363. Use the weaker hbnaev 2057 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 2422 . 2 (∀𝑥 𝑥 = 𝑦 → ∀𝑧𝑥 𝑥 = 𝑦)
21hbn 2283 1 (¬ ∀𝑥 𝑥 = 𝑦 → ∀𝑧 ¬ ∀𝑥 𝑥 = 𝑦)
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wal 1531
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1905  ax-6 1963  ax-7 2003  ax-10 2129  ax-11 2146  ax-12 2163  ax-13 2363
This theorem depends on definitions:  df-bi 206  df-an 396  df-or 845  df-tru 1536  df-ex 1774  df-nf 1778
This theorem is referenced by:  hbnaes  2426  eujustALT  2558  bj-hbnaeb  36198  ax6e2nd  43868  ax6e2ndVD  44218  ax6e2ndeqVD  44219  ax6e2ndALT  44240  ax6e2ndeqALT  44241
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