Theorem nfeqf2 2375

Description: An equation between setvar is free of any other setvar. Usage of this theorem is discouraged because it depends on ax-13 2370. (Contributed by Wolf Lammen, 9-Jun-2019.) Remove dependency on ax-12 2170. (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 1828	. 2 ⊢ (∃𝑥 ¬ 𝑥 = 𝑦 ↔ ¬ ∀𝑥 𝑥 = 𝑦)
2		hbe1 2138	. . . . 5 ⊢ (∃𝑥 𝑧 = 𝑦 → ∀𝑥∃𝑥 𝑧 = 𝑦)
3		ax13lem2 2374	. . . . . 6 ⊢ (¬ 𝑥 = 𝑦 → (∃𝑥 𝑧 = 𝑦 → 𝑧 = 𝑦))
4		ax13lem1 2372	. . . . . 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 1866	. . . 4 ⊢ (∃𝑥 𝑧 = 𝑦 → (∃𝑥 ¬ 𝑥 = 𝑦 → ∃𝑥∀𝑥 𝑧 = 𝑦))
7		hbe1a 2139	. . . 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 1791	. 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 1538  ∃wex 1780  Ⅎwnf 1784

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1912  ax-6 1970  ax-7 2010  ax-10 2136  ax-13 2370

This theorem depends on definitions:  df-bi 206  df-an 396  df-ex 1781  df-nf 1785

This theorem is referenced by:  dveeq2  2376  nfeqf1  2377  sb4b  2473  sbal1  2526  copsexg  5491  axrepndlem1  10593  axpowndlem2  10599  axpowndlem3  10600  bj-dvelimdv  36194  bj-dvelimdv1  36195  wl-equsb3  36885  wl-sbcom2d-lem1  36888  wl-mo2df  36899  wl-eudf  36901  wl-euequf  36903