Theorem nfeqf 2398

Description: A variable is effectively not free in an equality if it is not either of the involved variables. Ⅎ version of ax-c9 36030. Usage of this theorem is discouraged because it depends on ax-13 2389. (Contributed by Mario Carneiro, 6-Oct-2016.) Remove dependency on ax-11 2160. (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 2155	. . 3 ⊢ Ⅎ𝑧 ¬ ∀𝑧 𝑧 = 𝑥
2		nfna1 2155	. . 3 ⊢ Ⅎ𝑧 ¬ ∀𝑧 𝑧 = 𝑦
3	1, 2	nfan 1899	. 2 ⊢ Ⅎ𝑧(¬ ∀𝑧 𝑧 = 𝑥 ∧ ¬ ∀𝑧 𝑧 = 𝑦)
4		equvinva 2036	. . 3 ⊢ (𝑥 = 𝑦 → ∃𝑤(𝑥 = 𝑤 ∧ 𝑦 = 𝑤))
5		dveeq1 2397	. . . . . . . 8 ⊢ (¬ ∀𝑧 𝑧 = 𝑥 → (𝑥 = 𝑤 → ∀𝑧 𝑥 = 𝑤))
6	5	imp 409	. . . . . . 7 ⊢ ((¬ ∀𝑧 𝑧 = 𝑥 ∧ 𝑥 = 𝑤) → ∀𝑧 𝑥 = 𝑤)
7		dveeq1 2397	. . . . . . . 8 ⊢ (¬ ∀𝑧 𝑧 = 𝑦 → (𝑦 = 𝑤 → ∀𝑧 𝑦 = 𝑤))
8	7	imp 409	. . . . . . 7 ⊢ ((¬ ∀𝑧 𝑧 = 𝑦 ∧ 𝑦 = 𝑤) → ∀𝑧 𝑦 = 𝑤)
9		equtr2 2033	. . . . . . . 8 ⊢ ((𝑥 = 𝑤 ∧ 𝑦 = 𝑤) → 𝑥 = 𝑦)
10	9	alanimi 1816	. . . . . . 7 ⊢ ((∀𝑧 𝑥 = 𝑤 ∧ ∀𝑧 𝑦 = 𝑤) → ∀𝑧 𝑥 = 𝑦)
11	6, 8, 10	syl2an 597	. . . . . 6 ⊢ (((¬ ∀𝑧 𝑧 = 𝑥 ∧ 𝑥 = 𝑤) ∧ (¬ ∀𝑧 𝑧 = 𝑦 ∧ 𝑦 = 𝑤)) → ∀𝑧 𝑥 = 𝑦)
12	11	an4s 658	. . . . 5 ⊢ (((¬ ∀𝑧 𝑧 = 𝑥 ∧ ¬ ∀𝑧 𝑧 = 𝑦) ∧ (𝑥 = 𝑤 ∧ 𝑦 = 𝑤)) → ∀𝑧 𝑥 = 𝑦)
13	12	ex 415	. . . 4 ⊢ ((¬ ∀𝑧 𝑧 = 𝑥 ∧ ¬ ∀𝑧 𝑧 = 𝑦) → ((𝑥 = 𝑤 ∧ 𝑦 = 𝑤) → ∀𝑧 𝑥 = 𝑦))
14	13	exlimdv 1933	. . 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 2291	1 ⊢ ((¬ ∀𝑧 𝑧 = 𝑥 ∧ ¬ ∀𝑧 𝑧 = 𝑦) → Ⅎ𝑧 𝑥 = 𝑦)

Syntax hints:  ¬ wn 3   → wi 4   ∧ wa 398  ∀wal 1534  ∃wex 1779  Ⅎwnf 1783

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1969  ax-7 2014  ax-10 2144  ax-12 2176  ax-13 2389

This theorem depends on definitions:  df-bi 209  df-an 399  df-or 844  df-tru 1539  df-ex 1780  df-nf 1784

This theorem is referenced by:  axc9  2399  dvelimf  2469  equvel  2478  2ax6elem  2492  wl-exeq  34778  wl-nfeqfb  34780  wl-equsb4  34797  wl-2sb6d  34798  wl-sbalnae  34802