Theorem hbnae 2432

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 2372. 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

Step	Hyp	Ref	Expression
1		hbae 2431	. 2 ⊢ (∀𝑥 𝑥 = 𝑦 → ∀𝑧∀𝑥 𝑥 = 𝑦)
2	1	hbn 2292	1 ⊢ (¬ ∀𝑥 𝑥 = 𝑦 → ∀𝑧 ¬ ∀𝑥 𝑥 = 𝑦)

Syntax hints:  ¬ wn 3   → wi 4  ∀wal 1537

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1913  ax-6 1971  ax-7 2011  ax-10 2137  ax-11 2154  ax-12 2171  ax-13 2372

This theorem depends on definitions:  df-bi 206  df-an 397  df-or 845  df-tru 1542  df-ex 1783  df-nf 1787

This theorem is referenced by:  hbnaes  2435  eujustALT  2572  bj-hbnaeb  35003  ax6e2nd  42178  ax6e2ndVD  42528  ax6e2ndeqVD  42529  ax6e2ndALT  42550  ax6e2ndeqALT  42551