Theorem nfeqf2 2333

Description: An equation between setvar is free of any other setvar. (Contributed by Wolf Lammen, 9-Jun-2019.)

Assertion

Ref	Expression
nfeqf2	⊢ (¬ ∀𝑥 𝑥 = 𝑦 → Ⅎ𝑥 𝑧 = 𝑦)

Distinct variable group:   𝑥,𝑧

Proof of Theorem nfeqf2

Step	Hyp	Ref	Expression
1		exnal 1794	. 2 ⊢ (∃𝑥 ¬ 𝑥 = 𝑦 ↔ ¬ ∀𝑥 𝑥 = 𝑦)
2		nfnf1 2071	. . 3 ⊢ Ⅎ𝑥Ⅎ𝑥 𝑧 = 𝑦
3		ax13lem2 2332	. . . . 5 ⊢ (¬ 𝑥 = 𝑦 → (∃𝑥 𝑧 = 𝑦 → 𝑧 = 𝑦))
4		ax13lem1 2284	. . . . 5 ⊢ (¬ 𝑥 = 𝑦 → (𝑧 = 𝑦 → ∀𝑥 𝑧 = 𝑦))
5	3, 4	syld 47	. . . 4 ⊢ (¬ 𝑥 = 𝑦 → (∃𝑥 𝑧 = 𝑦 → ∀𝑥 𝑧 = 𝑦))
6		df-nf 1750	. . . 4 ⊢ (Ⅎ𝑥 𝑧 = 𝑦 ↔ (∃𝑥 𝑧 = 𝑦 → ∀𝑥 𝑧 = 𝑦))
7	5, 6	sylibr 224	. . 3 ⊢ (¬ 𝑥 = 𝑦 → Ⅎ𝑥 𝑧 = 𝑦)
8	2, 7	exlimi 2124	. 2 ⊢ (∃𝑥 ¬ 𝑥 = 𝑦 → Ⅎ𝑥 𝑧 = 𝑦)
9	1, 8	sylbir 225	1 ⊢ (¬ ∀𝑥 𝑥 = 𝑦 → Ⅎ𝑥 𝑧 = 𝑦)

Syntax hints:  ¬ wn 3   → wi 4  ∀wal 1521  ∃wex 1744  Ⅎwnf 1748

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1762  ax-4 1777  ax-5 1879  ax-6 1945  ax-7 1981  ax-10 2059  ax-12 2087  ax-13 2282

This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-ex 1745  df-nf 1750

This theorem is referenced by:  dveeq2  2334  nfeqf1  2335  sbal1  2488  copsexg  4985  axrepndlem1  9452  axpowndlem2  9458  axpowndlem3  9459  bj-dvelimdv  32959  bj-dvelimdv1  32960  wl-equsb3  33467  wl-sbcom2d-lem1  33472  wl-mo2df  33482  wl-eudf  33484  wl-euequ1f  33486