Theorem axextnd 10002

Description: A version of the Axiom of Extensionality with no distinct variable conditions. Usage of this theorem is discouraged because it depends on ax-13 2379. (Contributed by NM, 14-Aug-2003.) (New usage is discouraged.)

Assertion

Ref	Expression
axextnd	⊢ ∃𝑥((𝑥 ∈ 𝑦 ↔ 𝑥 ∈ 𝑧) → 𝑦 = 𝑧)

Proof of Theorem axextnd

Dummy variable 𝑤 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		nfnae 2445	. . . . . . . 8 ⊢ Ⅎ𝑥 ¬ ∀𝑥 𝑥 = 𝑦
2		nfnae 2445	. . . . . . . 8 ⊢ Ⅎ𝑥 ¬ ∀𝑥 𝑥 = 𝑧
3	1, 2	nfan 1900	. . . . . . 7 ⊢ Ⅎ𝑥(¬ ∀𝑥 𝑥 = 𝑦 ∧ ¬ ∀𝑥 𝑥 = 𝑧)
4		nfcvf 2981	. . . . . . . . . 10 ⊢ (¬ ∀𝑥 𝑥 = 𝑦 → Ⅎ𝑥𝑦)
5	4	adantr 484	. . . . . . . . 9 ⊢ ((¬ ∀𝑥 𝑥 = 𝑦 ∧ ¬ ∀𝑥 𝑥 = 𝑧) → Ⅎ𝑥𝑦)
6	5	nfcrd 2945	. . . . . . . 8 ⊢ ((¬ ∀𝑥 𝑥 = 𝑦 ∧ ¬ ∀𝑥 𝑥 = 𝑧) → Ⅎ𝑥 𝑤 ∈ 𝑦)
7		nfcvf 2981	. . . . . . . . . 10 ⊢ (¬ ∀𝑥 𝑥 = 𝑧 → Ⅎ𝑥𝑧)
8	7	adantl 485	. . . . . . . . 9 ⊢ ((¬ ∀𝑥 𝑥 = 𝑦 ∧ ¬ ∀𝑥 𝑥 = 𝑧) → Ⅎ𝑥𝑧)
9	8	nfcrd 2945	. . . . . . . 8 ⊢ ((¬ ∀𝑥 𝑥 = 𝑦 ∧ ¬ ∀𝑥 𝑥 = 𝑧) → Ⅎ𝑥 𝑤 ∈ 𝑧)
10	6, 9	nfbid 1903	. . . . . . 7 ⊢ ((¬ ∀𝑥 𝑥 = 𝑦 ∧ ¬ ∀𝑥 𝑥 = 𝑧) → Ⅎ𝑥(𝑤 ∈ 𝑦 ↔ 𝑤 ∈ 𝑧))
11		elequ1 2118	. . . . . . . . 9 ⊢ (𝑤 = 𝑥 → (𝑤 ∈ 𝑦 ↔ 𝑥 ∈ 𝑦))
12		elequ1 2118	. . . . . . . . 9 ⊢ (𝑤 = 𝑥 → (𝑤 ∈ 𝑧 ↔ 𝑥 ∈ 𝑧))
13	11, 12	bibi12d 349	. . . . . . . 8 ⊢ (𝑤 = 𝑥 → ((𝑤 ∈ 𝑦 ↔ 𝑤 ∈ 𝑧) ↔ (𝑥 ∈ 𝑦 ↔ 𝑥 ∈ 𝑧)))
14	13	a1i 11	. . . . . . 7 ⊢ ((¬ ∀𝑥 𝑥 = 𝑦 ∧ ¬ ∀𝑥 𝑥 = 𝑧) → (𝑤 = 𝑥 → ((𝑤 ∈ 𝑦 ↔ 𝑤 ∈ 𝑧) ↔ (𝑥 ∈ 𝑦 ↔ 𝑥 ∈ 𝑧))))
15	3, 10, 14	cbvald 2417	. . . . . 6 ⊢ ((¬ ∀𝑥 𝑥 = 𝑦 ∧ ¬ ∀𝑥 𝑥 = 𝑧) → (∀𝑤(𝑤 ∈ 𝑦 ↔ 𝑤 ∈ 𝑧) ↔ ∀𝑥(𝑥 ∈ 𝑦 ↔ 𝑥 ∈ 𝑧)))
16		axextg 2772	. . . . . 6 ⊢ (∀𝑤(𝑤 ∈ 𝑦 ↔ 𝑤 ∈ 𝑧) → 𝑦 = 𝑧)
17	15, 16	syl6bir 257	. . . . 5 ⊢ ((¬ ∀𝑥 𝑥 = 𝑦 ∧ ¬ ∀𝑥 𝑥 = 𝑧) → (∀𝑥(𝑥 ∈ 𝑦 ↔ 𝑥 ∈ 𝑧) → 𝑦 = 𝑧))
18		19.8a 2178	. . . . 5 ⊢ (𝑦 = 𝑧 → ∃𝑥 𝑦 = 𝑧)
19	17, 18	syl6 35	. . . 4 ⊢ ((¬ ∀𝑥 𝑥 = 𝑦 ∧ ¬ ∀𝑥 𝑥 = 𝑧) → (∀𝑥(𝑥 ∈ 𝑦 ↔ 𝑥 ∈ 𝑧) → ∃𝑥 𝑦 = 𝑧))
20	19	ex 416	. . 3 ⊢ (¬ ∀𝑥 𝑥 = 𝑦 → (¬ ∀𝑥 𝑥 = 𝑧 → (∀𝑥(𝑥 ∈ 𝑦 ↔ 𝑥 ∈ 𝑧) → ∃𝑥 𝑦 = 𝑧)))
21		ax6e 2390	. . . . 5 ⊢ ∃𝑥 𝑥 = 𝑧
22		ax7 2023	. . . . . 6 ⊢ (𝑥 = 𝑦 → (𝑥 = 𝑧 → 𝑦 = 𝑧))
23	22	aleximi 1833	. . . . 5 ⊢ (∀𝑥 𝑥 = 𝑦 → (∃𝑥 𝑥 = 𝑧 → ∃𝑥 𝑦 = 𝑧))
24	21, 23	mpi 20	. . . 4 ⊢ (∀𝑥 𝑥 = 𝑦 → ∃𝑥 𝑦 = 𝑧)
25	24	a1d 25	. . 3 ⊢ (∀𝑥 𝑥 = 𝑦 → (∀𝑥(𝑥 ∈ 𝑦 ↔ 𝑥 ∈ 𝑧) → ∃𝑥 𝑦 = 𝑧))
26		ax6e 2390	. . . . 5 ⊢ ∃𝑥 𝑥 = 𝑦
27		ax7 2023	. . . . . . 7 ⊢ (𝑥 = 𝑧 → (𝑥 = 𝑦 → 𝑧 = 𝑦))
28		equcomi 2024	. . . . . . 7 ⊢ (𝑧 = 𝑦 → 𝑦 = 𝑧)
29	27, 28	syl6 35	. . . . . 6 ⊢ (𝑥 = 𝑧 → (𝑥 = 𝑦 → 𝑦 = 𝑧))
30	29	aleximi 1833	. . . . 5 ⊢ (∀𝑥 𝑥 = 𝑧 → (∃𝑥 𝑥 = 𝑦 → ∃𝑥 𝑦 = 𝑧))
31	26, 30	mpi 20	. . . 4 ⊢ (∀𝑥 𝑥 = 𝑧 → ∃𝑥 𝑦 = 𝑧)
32	31	a1d 25	. . 3 ⊢ (∀𝑥 𝑥 = 𝑧 → (∀𝑥(𝑥 ∈ 𝑦 ↔ 𝑥 ∈ 𝑧) → ∃𝑥 𝑦 = 𝑧))
33	20, 25, 32	pm2.61ii 186	. 2 ⊢ (∀𝑥(𝑥 ∈ 𝑦 ↔ 𝑥 ∈ 𝑧) → ∃𝑥 𝑦 = 𝑧)
34	33	19.35ri 1880	1 ⊢ ∃𝑥((𝑥 ∈ 𝑦 ↔ 𝑥 ∈ 𝑧) → 𝑦 = 𝑧)

Syntax hints:  ¬ wn 3   → wi 4   ↔ wb 209   ∧ wa 399  ∀wal 1536  ∃wex 1781  Ⅎwnfc 2936

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1911  ax-6 1970  ax-7 2015  ax-8 2113  ax-9 2121  ax-10 2142  ax-11 2158  ax-12 2175  ax-13 2379  ax-ext 2770

This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-tru 1541  df-ex 1782  df-nf 1786  df-clel 2870  df-nfc 2938

This theorem is referenced by:  zfcndext  10024  axextprim  33040  axextdfeq  33155  axextndbi  33162