Theorem hbnae-o 35003

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 2454 using ax-c11 34962. (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 34978	. 2 ⊢ (∀𝑥 𝑥 = 𝑦 → ∀𝑧∀𝑥 𝑥 = 𝑦)
2	1	hbn 2329	1 ⊢ (¬ ∀𝑥 𝑥 = 𝑦 → ∀𝑧 ¬ ∀𝑥 𝑥 = 𝑦)

Syntax hints:  ¬ wn 3   → wi 4  ∀wal 1656

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1896  ax-4 1910  ax-5 2011  ax-6 2077  ax-7 2114  ax-10 2194  ax-11 2209  ax-12 2222  ax-c5 34958  ax-c4 34959  ax-c7 34960  ax-c10 34961  ax-c11 34962  ax-c9 34965

This theorem depends on definitions:  df-bi 199  df-an 387  df-or 881  df-ex 1881  df-nf 1885

This theorem is referenced by:  dvelimf-o  35004  ax12indalem  35020  ax12inda2ALT  35021