Theorem hbnaev 2058

Description: Any variable is free in ¬ ∀𝑥𝑥 = 𝑦, if 𝑥 and 𝑦 are distinct. This condition is dropped in hbnae 2427, at the expense of more axiom dependencies. Instance of naev2 2057. (Contributed by NM, 13-May-1993.) (Revised by Wolf Lammen, 9-Apr-2021.)

Assertion

Ref	Expression
hbnaev	⊢ (¬ ∀𝑥 𝑥 = 𝑦 → ∀𝑧 ¬ ∀𝑥 𝑥 = 𝑦)

Distinct variable group:   𝑥,𝑦

Proof of Theorem hbnaev

Step	Hyp	Ref	Expression
1		naev2 2057	1 ⊢ (¬ ∀𝑥 𝑥 = 𝑦 → ∀𝑧 ¬ ∀𝑥 𝑥 = 𝑦)

Syntax hints:  ¬ wn 3   → wi 4  ∀wal 1532

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1790  ax-4 1804  ax-5 1906  ax-6 1964  ax-7 2004

This theorem depends on definitions:  df-bi 206  df-an 396  df-ex 1775

This theorem is referenced by:  nfnaew  2138  euae  2651