Theorem hbnae-o 39421

Description: All variables are effectively bound in a distinct variable specifier. Lemma L19 in [Megill] p. 446 (p. 14 of the preprint). Version of hbnae 2440 using ax-c11 39380. (Contributed by NM, 13-May-1993.) (Proof modification is discouraged.) (New usage is discouraged.)

Assertion

Ref	Expression
hbnae-o	⊢ (¬ ∀𝑥 𝑥 = 𝑦 → ∀𝑧 ¬ ∀𝑥 𝑥 = 𝑦)

Proof of Theorem hbnae-o

Step	Hyp	Ref	Expression
1		hbae-o 39396	. 2 ⊢ (∀𝑥 𝑥 = 𝑦 → ∀𝑧∀𝑥 𝑥 = 𝑦)
2	1	hbn 2306	1 ⊢ (¬ ∀𝑥 𝑥 = 𝑦 → ∀𝑧 ¬ ∀𝑥 𝑥 = 𝑦)

Syntax hints:  ¬ wn 3   → wi 4  ∀wal 1545

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1802  ax-4 1816  ax-5 1917  ax-6 1974  ax-7 2015  ax-10 2152  ax-11 2168  ax-12 2189  ax-c5 39376  ax-c4 39377  ax-c7 39378  ax-c10 39379  ax-c11 39380  ax-c9 39383

This theorem depends on definitions:  df-bi 208  df-an 397  df-or 854  df-ex 1787  df-nf 1791

This theorem is referenced by:  dvelimf-o  39422  ax12indalem  39438  ax12inda2ALT  39439