Theorem nfeqf2 2408

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, 9-Jun-2019.) Remove dependency on ax-12 2215. (Revised by Wolf Lammen, 16-Dec-2022.) (New usage is discouraged.)

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 1860	. 2 ⊢ (∃𝑥 ¬ 𝑥 = 𝑦 ↔ ¬ ∀𝑥 𝑥 = 𝑦)
2		hbe1 2180	. . . . 5 ⊢ (∃𝑥 𝑧 = 𝑦 → ∀𝑥∃𝑥 𝑧 = 𝑦)
3		ax13lem2 2407	. . . . . 6 ⊢ (¬ 𝑥 = 𝑦 → (∃𝑥 𝑧 = 𝑦 → 𝑧 = 𝑦))
4		ax13lem1 2405	. . . . . 6 ⊢ (¬ 𝑥 = 𝑦 → (𝑧 = 𝑦 → ∀𝑥 𝑧 = 𝑦))
5	3, 4	syldc 49	. . . . 5 ⊢ (∃𝑥 𝑧 = 𝑦 → (¬ 𝑥 = 𝑦 → ∀𝑥 𝑧 = 𝑦))
6	2, 5	eximdh 1897	. . . 4 ⊢ (∃𝑥 𝑧 = 𝑦 → (∃𝑥 ¬ 𝑥 = 𝑦 → ∃𝑥∀𝑥 𝑧 = 𝑦))
7		hbe1a 2181	. . . 4 ⊢ (∃𝑥∀𝑥 𝑧 = 𝑦 → ∀𝑥 𝑧 = 𝑦)
8	6, 7	syl6com 38	. . 3 ⊢ (∃𝑥 ¬ 𝑥 = 𝑦 → (∃𝑥 𝑧 = 𝑦 → ∀𝑥 𝑧 = 𝑦))
9	8	nfd 1823	. 2 ⊢ (∃𝑥 ¬ 𝑥 = 𝑦 → Ⅎ𝑥 𝑧 = 𝑦)
10	1, 9	sylbir 238	1 ⊢ (¬ ∀𝑥 𝑥 = 𝑦 → Ⅎ𝑥 𝑧 = 𝑦)

This proof depends on syntax axioms:  ¬ wn 3   → wi 4  ∀wal 1568  ∃wex 1812  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:  dveeq2  2409  nfeqf1  2410  sb4b  2506  sbal1  2559  copsexg  5472  axrepndlem1  10604  axpowndlem2  10610  axpowndlem3  10611  axtcond  37084  mh-setindnd  37143  bj-dvelimdv  37581  bj-dvelimdv1  37582  wl-equsb3  38306  wl-sbcom2d-lem1  38309  wl-mo2df  38320  wl-eudf  38322  wl-euequf  38324