Theorem nfnaew 2145

Description: All variables are effectively bound in a distinct variable specifier. Version of nfnae 2433 with a disjoint variable condition, which does not require ax-13 2371. (Contributed by Mario Carneiro, 11-Aug-2016.) Avoid ax-13 2371. (Revised by Gino Giotto, 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 2065	. 2 ⊢ (¬ ∀𝑥 𝑥 = 𝑦 → ∀𝑧 ¬ ∀𝑥 𝑥 = 𝑦)
2	1	nf5i 2142	1 ⊢ Ⅎ𝑧 ¬ ∀𝑥 𝑥 = 𝑦

Syntax hints:  ¬ wn 3  ∀wal 1539  Ⅎwnf 1785

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1913  ax-6 1971  ax-7 2011  ax-10 2137

This theorem depends on definitions:  df-bi 206  df-an 397  df-ex 1782  df-nf 1786

This theorem is referenced by:  nfriotadw  7375