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Theorem hbnae 2455
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 2391. Use the weaker hbnaev 2067 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 2454 . 2 (∀𝑥 𝑥 = 𝑦 → ∀𝑧𝑥 𝑥 = 𝑦)
21hbn 2304 1 (¬ ∀𝑥 𝑥 = 𝑦 → ∀𝑧 ¬ ∀𝑥 𝑥 = 𝑦)
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wal 1536
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 1911  ax-6 1970  ax-7 2015  ax-10 2145  ax-11 2161  ax-12 2178  ax-13 2391
This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-tru 1541  df-ex 1782  df-nf 1786
This theorem is referenced by:  hbnaes  2458  eujustALT  2656  bj-hbnaeb  34219  ax6e2nd  41198  ax6e2ndVD  41548  ax6e2ndeqVD  41549  ax6e2ndALT  41570  ax6e2ndeqALT  41571
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