Theorem nfeqf1 2410

Description: An equation between setvar is free of any other setvar. Usage of this theorem is discouraged because it depends on ax-13 2403. (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 2408	. 2 ⊢ (¬ ∀𝑥 𝑥 = 𝑦 → Ⅎ𝑥 𝑧 = 𝑦)
2		equcom 2051	. . 3 ⊢ (𝑧 = 𝑦 ↔ 𝑦 = 𝑧)
3	2	nfbii 1885	. 2 ⊢ (Ⅎ𝑥 𝑧 = 𝑦 ↔ Ⅎ𝑥 𝑦 = 𝑧)
4	1, 3	sylib 221	1 ⊢ (¬ ∀𝑥 𝑥 = 𝑦 → Ⅎ𝑥 𝑦 = 𝑧)

This proof depends on syntax axioms:  ¬ wn 3   → wi 4  ∀wal 1568  Ⅎwnf 1816

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1828  ax-4 1842  ax-5 1943  ax-6 2000  ax-7 2041  ax-10 2178  ax-13 2403

This proof depends on definitions:  df-bi 210  df-an 402  df-ex 1813  df-nf 1817

This theorem is used by:  dveeq1  2411  sbal2  2560  nfmod2  2585  nfiotad  6498  mh-setindnd  37143  wl-mo2df  38320  wl-eudf  38322