Theorem nfeqf1 2412

Description: An equation between setvar is free of any other setvar. Usage of this theorem is discouraged because it depends on ax-13 2405. (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 2410	. 2 ⊢ (¬ ∀𝑥 𝑥 = 𝑦 → Ⅎ𝑥 𝑧 = 𝑦)
2		equcom 2040	. . 3 ⊢ (𝑧 = 𝑦 ↔ 𝑦 = 𝑧)
3	2	nfbii 1874	. 2 ⊢ (Ⅎ𝑥 𝑧 = 𝑦 ↔ Ⅎ𝑥 𝑦 = 𝑧)
4	1, 3	sylib 220	1 ⊢ (¬ ∀𝑥 𝑥 = 𝑦 → Ⅎ𝑥 𝑦 = 𝑧)

Syntax hints:  ¬ wn 3   → wi 4  ∀wal 1560  Ⅎwnf 1805

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1817  ax-4 1831  ax-5 1932  ax-6 1989  ax-7 2030  ax-10 2177  ax-13 2405

This theorem depends on definitions:  df-bi 209  df-an 400  df-ex 1802  df-nf 1806

This theorem is referenced by:  dveeq1  2413  sbal2  2562  nfmod2  2587  nfiotad  6484  mh-setindnd  36902  wl-mo2df  38078  wl-eudf  38080