Theorem nfnaew 2185

Description: All variables are effectively bound in a distinct variable specifier. Version of nfnae 2463 with a disjoint variable condition, which does not require ax-13 2401. (Contributed by Mario Carneiro, 11-Aug-2016.) Avoid ax-13 2401. (Revised by GG, 10-Jan-2024.) (Proof shortened by Wolf Lammen, 25-Sep-2024.)

Assertion

Ref	Expression
nfnaew	⊢ Ⅎ𝑧 ¬ ∀𝑥 𝑥 = 𝑦

Distinct variable group:   𝑥,𝑦

Proof of Theorem nfnaew

Step	Hyp	Ref	Expression
1		hbnaev 2096	. 2 ⊢ (¬ ∀𝑥 𝑥 = 𝑦 → ∀𝑧 ¬ ∀𝑥 𝑥 = 𝑦)
2	1	nf5i 2182	1 ⊢ Ⅎ𝑧 ¬ ∀𝑥 𝑥 = 𝑦

This proof depends on syntax axioms:  ¬ wn 3  ∀wal 1567  Ⅎwnf 1815

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1827  ax-4 1841  ax-5 1942  ax-6 1999  ax-7 2040  ax-10 2177

This proof depends on definitions:  df-bi 210  df-an 402  df-ex 1812  df-nf 1816

This theorem is used by:  nfriotadw  7389