Theorem nfeqf1 2287

Description: An equation between setvar is free of any other setvar. (Contributed by Wolf Lammen, 10-Jun-2019.)

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 2285	. 2 ⊢ (¬ ∀𝑥 𝑥 = 𝑦 → Ⅎ𝑥 𝑧 = 𝑦)
2		equcom 1932	. . 3 ⊢ (𝑧 = 𝑦 ↔ 𝑦 = 𝑧)
3	2	nfbii 1770	. 2 ⊢ (Ⅎ𝑥 𝑧 = 𝑦 ↔ Ⅎ𝑥 𝑦 = 𝑧)
4	1, 3	sylib 207	1 ⊢ (¬ ∀𝑥 𝑥 = 𝑦 → Ⅎ𝑥 𝑦 = 𝑧)

Syntax hints:  ¬ wn 3   → wi 4  ∀wal 1473  Ⅎwnf 1699

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1713  ax-4 1728  ax-5 1827  ax-6 1875  ax-7 1922  ax-10 2006  ax-12 2034  ax-13 2234

This theorem depends on definitions:  df-bi 196  df-or 384  df-an 385  df-tru 1478  df-ex 1696  df-nf 1701

This theorem is referenced by:  dveeq1  2288  sbal2  2449  nfeud2  2470  wl-mo2df  32531  wl-eudf  32533