MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  hbnae Structured version   Visualization version   GIF version

Theorem hbnae 2446
Description: All variables are effectively bound in a distinct variable specifier. A version with a distinct variable condition based on fewer axioms is hbnaev 2058. Lemma L19 in [Megill] p. 446 (p. 14 of the preprint). (Contributed by NM, 13-May-1993.)
Assertion
Ref Expression
hbnae (¬ ∀𝑥 𝑥 = 𝑦 → ∀𝑧 ¬ ∀𝑥 𝑥 = 𝑦)

Proof of Theorem hbnae
StepHypRef Expression
1 hbae 2445 . 2 (∀𝑥 𝑥 = 𝑦 → ∀𝑧𝑥 𝑥 = 𝑦)
21hbn 2294 1 (¬ ∀𝑥 𝑥 = 𝑦 → ∀𝑧 ¬ ∀𝑥 𝑥 = 𝑦)
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wal 1526
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1787  ax-4 1801  ax-5 1902  ax-6 1961  ax-7 2006  ax-10 2136  ax-11 2151  ax-12 2167  ax-13 2381
This theorem depends on definitions:  df-bi 208  df-an 397  df-or 842  df-tru 1531  df-ex 1772  df-nf 1776
This theorem is referenced by:  hbnaes  2449  eujustALT  2650  bj-hbnaeb  34040  ax6e2nd  40769  ax6e2ndVD  41119  ax6e2ndeqVD  41120  ax6e2ndALT  41141  ax6e2ndeqALT  41142
  Copyright terms: Public domain W3C validator