Theorem nfeqf 2301

Description: A variable is effectively not free in an equality if it is not either of the involved variables. Ⅎ version of ax-c9 34175. (Contributed by Mario Carneiro, 6-Oct-2016.) Remove dependency on ax-11 2034. (Revised by Wolf Lammen, 6-Sep-2018.)

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 2029	. . 3 ⊢ Ⅎ𝑧 ¬ ∀𝑧 𝑧 = 𝑥
2		nfna1 2029	. . 3 ⊢ Ⅎ𝑧 ¬ ∀𝑧 𝑧 = 𝑦
3	1, 2	nfan 1828	. 2 ⊢ Ⅎ𝑧(¬ ∀𝑧 𝑧 = 𝑥 ∧ ¬ ∀𝑧 𝑧 = 𝑦)
4		equviniva 1960	. . 3 ⊢ (𝑥 = 𝑦 → ∃𝑤(𝑥 = 𝑤 ∧ 𝑦 = 𝑤))
5		dveeq1 2300	. . . . . . . 8 ⊢ (¬ ∀𝑧 𝑧 = 𝑥 → (𝑥 = 𝑤 → ∀𝑧 𝑥 = 𝑤))
6	5	imp 445	. . . . . . 7 ⊢ ((¬ ∀𝑧 𝑧 = 𝑥 ∧ 𝑥 = 𝑤) → ∀𝑧 𝑥 = 𝑤)
7		dveeq1 2300	. . . . . . . 8 ⊢ (¬ ∀𝑧 𝑧 = 𝑦 → (𝑦 = 𝑤 → ∀𝑧 𝑦 = 𝑤))
8	7	imp 445	. . . . . . 7 ⊢ ((¬ ∀𝑧 𝑧 = 𝑦 ∧ 𝑦 = 𝑤) → ∀𝑧 𝑦 = 𝑤)
9		equtr2 1954	. . . . . . . 8 ⊢ ((𝑥 = 𝑤 ∧ 𝑦 = 𝑤) → 𝑥 = 𝑦)
10	9	alanimi 1744	. . . . . . 7 ⊢ ((∀𝑧 𝑥 = 𝑤 ∧ ∀𝑧 𝑦 = 𝑤) → ∀𝑧 𝑥 = 𝑦)
11	6, 8, 10	syl2an 494	. . . . . 6 ⊢ (((¬ ∀𝑧 𝑧 = 𝑥 ∧ 𝑥 = 𝑤) ∧ (¬ ∀𝑧 𝑧 = 𝑦 ∧ 𝑦 = 𝑤)) → ∀𝑧 𝑥 = 𝑦)
12	11	an4s 869	. . . . 5 ⊢ (((¬ ∀𝑧 𝑧 = 𝑥 ∧ ¬ ∀𝑧 𝑧 = 𝑦) ∧ (𝑥 = 𝑤 ∧ 𝑦 = 𝑤)) → ∀𝑧 𝑥 = 𝑦)
13	12	ex 450	. . . 4 ⊢ ((¬ ∀𝑧 𝑧 = 𝑥 ∧ ¬ ∀𝑧 𝑧 = 𝑦) → ((𝑥 = 𝑤 ∧ 𝑦 = 𝑤) → ∀𝑧 𝑥 = 𝑦))
14	13	exlimdv 1861	. . 3 ⊢ ((¬ ∀𝑧 𝑧 = 𝑥 ∧ ¬ ∀𝑧 𝑧 = 𝑦) → (∃𝑤(𝑥 = 𝑤 ∧ 𝑦 = 𝑤) → ∀𝑧 𝑥 = 𝑦))
15	4, 14	syl5 34	. 2 ⊢ ((¬ ∀𝑧 𝑧 = 𝑥 ∧ ¬ ∀𝑧 𝑧 = 𝑦) → (𝑥 = 𝑦 → ∀𝑧 𝑥 = 𝑦))
16	3, 15	nf5d 2118	1 ⊢ ((¬ ∀𝑧 𝑧 = 𝑥 ∧ ¬ ∀𝑧 𝑧 = 𝑦) → Ⅎ𝑧 𝑥 = 𝑦)

Syntax hints:  ¬ wn 3   → wi 4   ∧ wa 384  ∀wal 1481  ∃wex 1704  Ⅎwnf 1708

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1722  ax-4 1737  ax-5 1839  ax-6 1888  ax-7 1935  ax-10 2019  ax-12 2047  ax-13 2246

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-tru 1486  df-ex 1705  df-nf 1710

This theorem is referenced by:  axc9  2302  dvelimf  2334  equvel  2347  2ax6elem  2449  wl-exeq  33321  wl-nfeqfb  33323  wl-equsb4  33338  wl-2sb6d  33341  wl-sbalnae  33345