Theorem nfeqf1 2386

Description: An equation between setvar is free of any other setvar. Usage of this theorem is discouraged because it depends on ax-13 2379. (Contributed by Wolf Lammen, 10-Jun-2019.) (New usage is discouraged.)

Assertion

Ref	Expression
nfeqf1	⊢ (¬ ∀𝑥 𝑥 = 𝑦 → Ⅎ𝑥 𝑦 = 𝑧)

Distinct variable group:   𝑥,𝑧

Proof of Theorem nfeqf1

Step	Hyp	Ref	Expression
1		nfeqf2 2384	. 2 ⊢ (¬ ∀𝑥 𝑥 = 𝑦 → Ⅎ𝑥 𝑧 = 𝑦)
2		equcom 2025	. . 3 ⊢ (𝑧 = 𝑦 ↔ 𝑦 = 𝑧)
3	2	nfbii 1853	. 2 ⊢ (Ⅎ𝑥 𝑧 = 𝑦 ↔ Ⅎ𝑥 𝑦 = 𝑧)
4	1, 3	sylib 221	1 ⊢ (¬ ∀𝑥 𝑥 = 𝑦 → Ⅎ𝑥 𝑦 = 𝑧)

Syntax hints:  ¬ wn 3   → wi 4  ∀wal 1536  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 1911  ax-6 1970  ax-7 2015  ax-10 2142  ax-13 2379

This theorem depends on definitions:  df-bi 210  df-an 400  df-ex 1782  df-nf 1786

This theorem is referenced by:  dveeq1  2387  sbal2  2549  sbal2OLD  2550  nfmod2  2617  nfiotad  6288  wl-mo2df  34971  wl-eudf  34973