Theorem hbnae 2465

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 2405. Use the weaker hbnaev 2086 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 2464	. 2 ⊢ (∀𝑥 𝑥 = 𝑦 → ∀𝑧∀𝑥 𝑥 = 𝑦)
2	1	hbn 2331	1 ⊢ (¬ ∀𝑥 𝑥 = 𝑦 → ∀𝑧 ¬ ∀𝑥 𝑥 = 𝑦)

Syntax hints:  ¬ wn 3   → wi 4  ∀wal 1560

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1817  ax-4 1831  ax-5 1932  ax-6 1989  ax-7 2030  ax-10 2177  ax-11 2193  ax-12 2214  ax-13 2405

This theorem depends on definitions:  df-bi 209  df-an 400  df-or 859  df-tru 1565  df-ex 1802  df-nf 1806

This theorem is referenced by:  hbnaes  2468  eujustALT  2601  bj-hbnaeb  37310  ax6e2nd  45139  ax6e2ndVD  45488  ax6e2ndeqVD  45489  ax6e2ndALT  45510  ax6e2ndeqALT  45511