Theorem nfeqf2 2285

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 1744	. 2 ⊢ (∃𝑥 ¬ 𝑥 = 𝑦 ↔ ¬ ∀𝑥 𝑥 = 𝑦)
2		nfnf1 2018	. . 3 ⊢ Ⅎ𝑥Ⅎ𝑥 𝑧 = 𝑦
3		ax13lem2 2284	. . . . 5 ⊢ (¬ 𝑥 = 𝑦 → (∃𝑥 𝑧 = 𝑦 → 𝑧 = 𝑦))
4		ax13lem1 2236	. . . . 5 ⊢ (¬ 𝑥 = 𝑦 → (𝑧 = 𝑦 → ∀𝑥 𝑧 = 𝑦))
5	3, 4	syld 46	. . . 4 ⊢ (¬ 𝑥 = 𝑦 → (∃𝑥 𝑧 = 𝑦 → ∀𝑥 𝑧 = 𝑦))
6		df-nf 1701	. . . 4 ⊢ (Ⅎ𝑥 𝑧 = 𝑦 ↔ (∃𝑥 𝑧 = 𝑦 → ∀𝑥 𝑧 = 𝑦))
7	5, 6	sylibr 223	. . 3 ⊢ (¬ 𝑥 = 𝑦 → Ⅎ𝑥 𝑧 = 𝑦)
8	2, 7	exlimi 2073	. 2 ⊢ (∃𝑥 ¬ 𝑥 = 𝑦 → Ⅎ𝑥 𝑧 = 𝑦)
9	1, 8	sylbir 224	1 ⊢ (¬ ∀𝑥 𝑥 = 𝑦 → Ⅎ𝑥 𝑧 = 𝑦)

Syntax hints:  ¬ wn 3   → wi 4  ∀wal 1473  ∃wex 1695  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:  dveeq2  2286  nfeqf1  2287  sbal1  2448  copsexg  4882  axrepndlem1  9293  axpowndlem2  9299  axpowndlem3  9300  bj-dvelimdv  32027  bj-dvelimdv1  32028  wl-equsb3  32516  wl-sbcom2d-lem1  32521  wl-mo2df  32531  wl-eudf  32533  wl-euequ1f  32535