Theorem nfeqf 2410

Description: A variable is effectively not free in an equality if it is not either of the involved variables. Ⅎ version of ax-c9 39990. Usage of this theorem is discouraged because it depends on ax-13 2401. (Contributed by Mario Carneiro, 6-Oct-2016.) Remove dependency on ax-11 2193. (Revised by Wolf Lammen, 6-Sep-2018.) (New usage is discouraged.)

Assertion

Ref	Expression
nfeqf	⊢ ((¬ ∀𝑧 𝑧 = 𝑥 ∧ ¬ ∀𝑧 𝑧 = 𝑦) → Ⅎ𝑧 𝑥 = 𝑦)

Proof of Theorem nfeqf

Dummy variable 𝑤 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		nfna1 2188	. . 3 ⊢ Ⅎ𝑧 ¬ ∀𝑧 𝑧 = 𝑥
2		nfna1 2188	. . 3 ⊢ Ⅎ𝑧 ¬ ∀𝑧 𝑧 = 𝑦
3	1, 2	nfan 1931	. 2 ⊢ Ⅎ𝑧(¬ ∀𝑧 𝑧 = 𝑥 ∧ ¬ ∀𝑧 𝑧 = 𝑦)
4		equvinva 2062	. . 3 ⊢ (𝑥 = 𝑦 → ∃𝑤(𝑥 = 𝑤 ∧ 𝑦 = 𝑤))
5		dveeq1 2409	. . . . . . . 8 ⊢ (¬ ∀𝑧 𝑧 = 𝑥 → (𝑥 = 𝑤 → ∀𝑧 𝑥 = 𝑤))
6	5	imp 412	. . . . . . 7 ⊢ ((¬ ∀𝑧 𝑧 = 𝑥 ∧ 𝑥 = 𝑤) → ∀𝑧 𝑥 = 𝑤)
7		dveeq1 2409	. . . . . . . 8 ⊢ (¬ ∀𝑧 𝑧 = 𝑦 → (𝑦 = 𝑤 → ∀𝑧 𝑦 = 𝑤))
8	7	imp 412	. . . . . . 7 ⊢ ((¬ ∀𝑧 𝑧 = 𝑦 ∧ 𝑦 = 𝑤) → ∀𝑧 𝑦 = 𝑤)
9		equtr2 2059	. . . . . . . 8 ⊢ ((𝑥 = 𝑤 ∧ 𝑦 = 𝑤) → 𝑥 = 𝑦)
10	9	alanimi 1848	. . . . . . 7 ⊢ ((∀𝑧 𝑥 = 𝑤 ∧ ∀𝑧 𝑦 = 𝑤) → ∀𝑧 𝑥 = 𝑦)
11	6, 8, 10	syl2an 608	. . . . . 6 ⊢ (((¬ ∀𝑧 𝑧 = 𝑥 ∧ 𝑥 = 𝑤) ∧ (¬ ∀𝑧 𝑧 = 𝑦 ∧ 𝑦 = 𝑤)) → ∀𝑧 𝑥 = 𝑦)
12	11	an4s 673	. . . . 5 ⊢ (((¬ ∀𝑧 𝑧 = 𝑥 ∧ ¬ ∀𝑧 𝑧 = 𝑦) ∧ (𝑥 = 𝑤 ∧ 𝑦 = 𝑤)) → ∀𝑧 𝑥 = 𝑦)
13	12	ex 418	. . . 4 ⊢ ((¬ ∀𝑧 𝑧 = 𝑥 ∧ ¬ ∀𝑧 𝑧 = 𝑦) → ((𝑥 = 𝑤 ∧ 𝑦 = 𝑤) → ∀𝑧 𝑥 = 𝑦))
14	13	exlimdv 1965	. . 3 ⊢ ((¬ ∀𝑧 𝑧 = 𝑥 ∧ ¬ ∀𝑧 𝑧 = 𝑦) → (∃𝑤(𝑥 = 𝑤 ∧ 𝑦 = 𝑤) → ∀𝑧 𝑥 = 𝑦))
15	4, 14	syl5 35	. 2 ⊢ ((¬ ∀𝑧 𝑧 = 𝑥 ∧ ¬ ∀𝑧 𝑧 = 𝑦) → (𝑥 = 𝑦 → ∀𝑧 𝑥 = 𝑦))
16	3, 15	nf5d 2317	1 ⊢ ((¬ ∀𝑧 𝑧 = 𝑥 ∧ ¬ ∀𝑧 𝑧 = 𝑦) → Ⅎ𝑧 𝑥 = 𝑦)

This proof depends on syntax axioms:  ¬ wn 3   → wi 4   ∧ wa 401  ∀wal 1567  ∃wex 1811  Ⅎwnf 1815

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1827  ax-4 1841  ax-5 1942  ax-6 1999  ax-7 2040  ax-10 2177  ax-12 2212  ax-13 2401

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-tru 1572  df-ex 1812  df-nf 1816

This theorem is used by:  axc9  2411  dvelimf  2477  equvel  2485  2ax6elem  2499  wl-exeq  38509  wl-nfeqfb  38511  wl-equsb4  38532  wl-2sb6d  38533  wl-sbalnae  38537