Theorem nfeqf2 2376

Description: An equation between setvar is free of any other setvar. Usage of this theorem is discouraged because it depends on ax-13 2371. (Contributed by Wolf Lammen, 9-Jun-2019.) Remove dependency on ax-12 2171. (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 1829	. 2 ⊢ (∃𝑥 ¬ 𝑥 = 𝑦 ↔ ¬ ∀𝑥 𝑥 = 𝑦)
2		hbe1 2139	. . . . 5 ⊢ (∃𝑥 𝑧 = 𝑦 → ∀𝑥∃𝑥 𝑧 = 𝑦)
3		ax13lem2 2375	. . . . . 6 ⊢ (¬ 𝑥 = 𝑦 → (∃𝑥 𝑧 = 𝑦 → 𝑧 = 𝑦))
4		ax13lem1 2373	. . . . . 6 ⊢ (¬ 𝑥 = 𝑦 → (𝑧 = 𝑦 → ∀𝑥 𝑧 = 𝑦))
5	3, 4	syldc 48	. . . . 5 ⊢ (∃𝑥 𝑧 = 𝑦 → (¬ 𝑥 = 𝑦 → ∀𝑥 𝑧 = 𝑦))
6	2, 5	eximdh 1867	. . . 4 ⊢ (∃𝑥 𝑧 = 𝑦 → (∃𝑥 ¬ 𝑥 = 𝑦 → ∃𝑥∀𝑥 𝑧 = 𝑦))
7		hbe1a 2140	. . . 4 ⊢ (∃𝑥∀𝑥 𝑧 = 𝑦 → ∀𝑥 𝑧 = 𝑦)
8	6, 7	syl6com 37	. . 3 ⊢ (∃𝑥 ¬ 𝑥 = 𝑦 → (∃𝑥 𝑧 = 𝑦 → ∀𝑥 𝑧 = 𝑦))
9	8	nfd 1792	. 2 ⊢ (∃𝑥 ¬ 𝑥 = 𝑦 → Ⅎ𝑥 𝑧 = 𝑦)
10	1, 9	sylbir 234	1 ⊢ (¬ ∀𝑥 𝑥 = 𝑦 → Ⅎ𝑥 𝑧 = 𝑦)

Syntax hints:  ¬ wn 3   → wi 4  ∀wal 1539  ∃wex 1781  Ⅎwnf 1785

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1913  ax-6 1971  ax-7 2011  ax-10 2137  ax-13 2371

This theorem depends on definitions:  df-bi 206  df-an 397  df-ex 1782  df-nf 1786

This theorem is referenced by:  dveeq2  2377  nfeqf1  2378  sb4b  2474  sbal1  2527  copsexg  5491  axrepndlem1  10586  axpowndlem2  10592  axpowndlem3  10593  bj-dvelimdv  35725  bj-dvelimdv1  35726  wl-equsb3  36416  wl-sbcom2d-lem1  36419  wl-mo2df  36430  wl-eudf  36432  wl-euequf  36434