Theorem nfeqf2 2387

Description: An equation between setvar is free of any other setvar. Usage of this theorem is discouraged because it depends on ax-13 2382. (Contributed by Wolf Lammen, 9-Jun-2019.) Remove dependency on ax-12 2191. (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 1835	. 2 ⊢ (∃𝑥 ¬ 𝑥 = 𝑦 ↔ ¬ ∀𝑥 𝑥 = 𝑦)
2		hbe1 2156	. . . . 5 ⊢ (∃𝑥 𝑧 = 𝑦 → ∀𝑥∃𝑥 𝑧 = 𝑦)
3		ax13lem2 2386	. . . . . 6 ⊢ (¬ 𝑥 = 𝑦 → (∃𝑥 𝑧 = 𝑦 → 𝑧 = 𝑦))
4		ax13lem1 2384	. . . . . 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 1872	. . . 4 ⊢ (∃𝑥 𝑧 = 𝑦 → (∃𝑥 ¬ 𝑥 = 𝑦 → ∃𝑥∀𝑥 𝑧 = 𝑦))
7		hbe1a 2157	. . . 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 1798	. 2 ⊢ (∃𝑥 ¬ 𝑥 = 𝑦 → Ⅎ𝑥 𝑧 = 𝑦)
10	1, 9	sylbir 237	1 ⊢ (¬ ∀𝑥 𝑥 = 𝑦 → Ⅎ𝑥 𝑧 = 𝑦)

Syntax hints:  ¬ wn 3   → wi 4  ∀wal 1546  ∃wex 1787  Ⅎwnf 1791

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1803  ax-4 1817  ax-5 1918  ax-6 1975  ax-7 2016  ax-10 2154  ax-13 2382

This theorem depends on definitions:  df-bi 209  df-an 398  df-ex 1788  df-nf 1792

This theorem is referenced by:  dveeq2  2388  nfeqf1  2389  sb4b  2485  sbal1  2538  copsexg  5435  axrepndlem1  10510  axpowndlem2  10516  axpowndlem3  10517  axtcond  36721  mh-setindnd  36780  bj-dvelimdv  37219  bj-dvelimdv1  37220  wl-equsb3  37942  wl-sbcom2d-lem1  37945  wl-mo2df  37956  wl-eudf  37958  wl-euequf  37960