Theorem axsepg2 35328

Description: A generalization of ax-sep 5225 in which 𝑥 and 𝑧 need not be distinct. This theorem scheme bundles ax-sep 5225 with the degenerate instance ∃𝑦∀𝑧(𝑧 ∈ 𝑦 ↔ (𝑧 ∈ 𝑧 ∧ 𝜑)) which is satisfied by the existence of the empty set. Usage of this theorem is discouraged because it depends on ax-13 2380. (Contributed by BTernaryTau, 21-May-2026.) (New usage is discouraged.)

Assertion

Ref	Expression
axsepg2	⊢ ∃𝑦∀𝑥(𝑥 ∈ 𝑦 ↔ (𝑥 ∈ 𝑧 ∧ 𝜑))

Distinct variable groups:   𝑥,𝑦   𝜑,𝑦,𝑧

Allowed substitution hint:   𝜑(𝑥)

Proof of Theorem axsepg2

Dummy variable 𝑤 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		nfv 1921	. . . 4 ⊢ Ⅎ𝑦 ¬ ∀𝑧 𝑧 = 𝑥
2		nfnae 2442	. . . . 5 ⊢ Ⅎ𝑥 ¬ ∀𝑧 𝑧 = 𝑥
3		nfcvf 2928	. . . . . . 7 ⊢ (¬ ∀𝑧 𝑧 = 𝑥 → Ⅎ𝑧𝑥)
4		nfcvd 2903	. . . . . . 7 ⊢ (¬ ∀𝑧 𝑧 = 𝑥 → Ⅎ𝑧𝑦)
5	3, 4	nfeld 2913	. . . . . 6 ⊢ (¬ ∀𝑧 𝑧 = 𝑥 → Ⅎ𝑧 𝑥 ∈ 𝑦)
6		nfcvd 2903	. . . . . . . 8 ⊢ (¬ ∀𝑧 𝑧 = 𝑥 → Ⅎ𝑧𝑤)
7	3, 6	nfeld 2913	. . . . . . 7 ⊢ (¬ ∀𝑧 𝑧 = 𝑥 → Ⅎ𝑧 𝑥 ∈ 𝑤)
8		nfvd 1922	. . . . . . 7 ⊢ (¬ ∀𝑧 𝑧 = 𝑥 → Ⅎ𝑧𝜑)
9	7, 8	nfand 1904	. . . . . 6 ⊢ (¬ ∀𝑧 𝑧 = 𝑥 → Ⅎ𝑧(𝑥 ∈ 𝑤 ∧ 𝜑))
10	5, 9	nfbid 1909	. . . . 5 ⊢ (¬ ∀𝑧 𝑧 = 𝑥 → Ⅎ𝑧(𝑥 ∈ 𝑦 ↔ (𝑥 ∈ 𝑤 ∧ 𝜑)))
11	2, 10	nfald 2337	. . . 4 ⊢ (¬ ∀𝑧 𝑧 = 𝑥 → Ⅎ𝑧∀𝑥(𝑥 ∈ 𝑦 ↔ (𝑥 ∈ 𝑤 ∧ 𝜑)))
12	1, 11	nfexd 2338	. . 3 ⊢ (¬ ∀𝑧 𝑧 = 𝑥 → Ⅎ𝑧∃𝑦∀𝑥(𝑥 ∈ 𝑦 ↔ (𝑥 ∈ 𝑤 ∧ 𝜑)))
13		nfvd 1922	. . 3 ⊢ (¬ ∀𝑧 𝑧 = 𝑥 → Ⅎ𝑤∃𝑦∀𝑥(𝑥 ∈ 𝑦 ↔ (𝑥 ∈ 𝑧 ∧ 𝜑)))
14		dveeq2 2386	. . . . 5 ⊢ (¬ ∀𝑥 𝑥 = 𝑧 → (𝑤 = 𝑧 → ∀𝑥 𝑤 = 𝑧))
15	14	naecoms 2437	. . . 4 ⊢ (¬ ∀𝑧 𝑧 = 𝑥 → (𝑤 = 𝑧 → ∀𝑥 𝑤 = 𝑧))
16		elequ2 2134	. . . . . . . . 9 ⊢ (𝑤 = 𝑧 → (𝑥 ∈ 𝑤 ↔ 𝑥 ∈ 𝑧))
17	16	anbi1d 637	. . . . . . . 8 ⊢ (𝑤 = 𝑧 → ((𝑥 ∈ 𝑤 ∧ 𝜑) ↔ (𝑥 ∈ 𝑧 ∧ 𝜑)))
18	17	bibi2d 343	. . . . . . 7 ⊢ (𝑤 = 𝑧 → ((𝑥 ∈ 𝑦 ↔ (𝑥 ∈ 𝑤 ∧ 𝜑)) ↔ (𝑥 ∈ 𝑦 ↔ (𝑥 ∈ 𝑧 ∧ 𝜑))))
19	18	biimpd 230	. . . . . 6 ⊢ (𝑤 = 𝑧 → ((𝑥 ∈ 𝑦 ↔ (𝑥 ∈ 𝑤 ∧ 𝜑)) → (𝑥 ∈ 𝑦 ↔ (𝑥 ∈ 𝑧 ∧ 𝜑))))
20	19	al2imi 1822	. . . . 5 ⊢ (∀𝑥 𝑤 = 𝑧 → (∀𝑥(𝑥 ∈ 𝑦 ↔ (𝑥 ∈ 𝑤 ∧ 𝜑)) → ∀𝑥(𝑥 ∈ 𝑦 ↔ (𝑥 ∈ 𝑧 ∧ 𝜑))))
21	20	eximdv 1924	. . . 4 ⊢ (∀𝑥 𝑤 = 𝑧 → (∃𝑦∀𝑥(𝑥 ∈ 𝑦 ↔ (𝑥 ∈ 𝑤 ∧ 𝜑)) → ∃𝑦∀𝑥(𝑥 ∈ 𝑦 ↔ (𝑥 ∈ 𝑧 ∧ 𝜑))))
22	15, 21	syl6 35	. . 3 ⊢ (¬ ∀𝑧 𝑧 = 𝑥 → (𝑤 = 𝑧 → (∃𝑦∀𝑥(𝑥 ∈ 𝑦 ↔ (𝑥 ∈ 𝑤 ∧ 𝜑)) → ∃𝑦∀𝑥(𝑥 ∈ 𝑦 ↔ (𝑥 ∈ 𝑧 ∧ 𝜑)))))
23		elequ1 2126	. . . . . . . 8 ⊢ (𝑧 = 𝑥 → (𝑧 ∈ 𝑦 ↔ 𝑥 ∈ 𝑦))
24		elequ1 2126	. . . . . . . . 9 ⊢ (𝑧 = 𝑥 → (𝑧 ∈ 𝑧 ↔ 𝑥 ∈ 𝑧))
25	24	anbi1d 637	. . . . . . . 8 ⊢ (𝑧 = 𝑥 → ((𝑧 ∈ 𝑧 ∧ 𝜑) ↔ (𝑥 ∈ 𝑧 ∧ 𝜑)))
26	23, 25	bibi12d 346	. . . . . . 7 ⊢ (𝑧 = 𝑥 → ((𝑧 ∈ 𝑦 ↔ (𝑧 ∈ 𝑧 ∧ 𝜑)) ↔ (𝑥 ∈ 𝑦 ↔ (𝑥 ∈ 𝑧 ∧ 𝜑))))
27	26	biimpd 230	. . . . . 6 ⊢ (𝑧 = 𝑥 → ((𝑧 ∈ 𝑦 ↔ (𝑧 ∈ 𝑧 ∧ 𝜑)) → (𝑥 ∈ 𝑦 ↔ (𝑥 ∈ 𝑧 ∧ 𝜑))))
28	27	al2imi 1822	. . . . 5 ⊢ (∀𝑧 𝑧 = 𝑥 → (∀𝑧(𝑧 ∈ 𝑦 ↔ (𝑧 ∈ 𝑧 ∧ 𝜑)) → ∀𝑧(𝑥 ∈ 𝑦 ↔ (𝑥 ∈ 𝑧 ∧ 𝜑))))
29		axc11 2438	. . . . 5 ⊢ (∀𝑧 𝑧 = 𝑥 → (∀𝑧(𝑥 ∈ 𝑦 ↔ (𝑥 ∈ 𝑧 ∧ 𝜑)) → ∀𝑥(𝑥 ∈ 𝑦 ↔ (𝑥 ∈ 𝑧 ∧ 𝜑))))
30	28, 29	syld 47	. . . 4 ⊢ (∀𝑧 𝑧 = 𝑥 → (∀𝑧(𝑧 ∈ 𝑦 ↔ (𝑧 ∈ 𝑧 ∧ 𝜑)) → ∀𝑥(𝑥 ∈ 𝑦 ↔ (𝑥 ∈ 𝑧 ∧ 𝜑))))
31	30	eximdv 1924	. . 3 ⊢ (∀𝑧 𝑧 = 𝑥 → (∃𝑦∀𝑧(𝑧 ∈ 𝑦 ↔ (𝑧 ∈ 𝑧 ∧ 𝜑)) → ∃𝑦∀𝑥(𝑥 ∈ 𝑦 ↔ (𝑥 ∈ 𝑧 ∧ 𝜑))))
32		ax-sep 5225	. . . 4 ⊢ ∃𝑦∀𝑥(𝑥 ∈ 𝑦 ↔ (𝑥 ∈ 𝑤 ∧ 𝜑))
33	32	ax-gen 1802	. . 3 ⊢ ∀𝑤∃𝑦∀𝑥(𝑥 ∈ 𝑦 ↔ (𝑥 ∈ 𝑤 ∧ 𝜑))
34		ax-nul 5235	. . . . 5 ⊢ ∃𝑦∀𝑧 ¬ 𝑧 ∈ 𝑦
35		elirrv 9509	. . . . . . . . 9 ⊢ ¬ 𝑧 ∈ 𝑧
36	35	intnanr 488	. . . . . . . 8 ⊢ ¬ (𝑧 ∈ 𝑧 ∧ 𝜑)
37	36	nbn 373	. . . . . . 7 ⊢ (¬ 𝑧 ∈ 𝑦 ↔ (𝑧 ∈ 𝑦 ↔ (𝑧 ∈ 𝑧 ∧ 𝜑)))
38	37	biimpi 217	. . . . . 6 ⊢ (¬ 𝑧 ∈ 𝑦 → (𝑧 ∈ 𝑦 ↔ (𝑧 ∈ 𝑧 ∧ 𝜑)))
39	38	alimi 1818	. . . . 5 ⊢ (∀𝑧 ¬ 𝑧 ∈ 𝑦 → ∀𝑧(𝑧 ∈ 𝑦 ↔ (𝑧 ∈ 𝑧 ∧ 𝜑)))
40	34, 39	eximii 1844	. . . 4 ⊢ ∃𝑦∀𝑧(𝑧 ∈ 𝑦 ↔ (𝑧 ∈ 𝑧 ∧ 𝜑))
41	40	ax-gen 1802	. . 3 ⊢ ∀𝑧∃𝑦∀𝑧(𝑧 ∈ 𝑦 ↔ (𝑧 ∈ 𝑧 ∧ 𝜑))
42	12, 13, 22, 31, 33, 41	dvelimalcasei 35265	. 2 ⊢ ∀𝑧∃𝑦∀𝑥(𝑥 ∈ 𝑦 ↔ (𝑥 ∈ 𝑧 ∧ 𝜑))
43	42	spi 2196	1 ⊢ ∃𝑦∀𝑥(𝑥 ∈ 𝑦 ↔ (𝑥 ∈ 𝑧 ∧ 𝜑))

Syntax hints:  ¬ wn 3   → wi 4   ↔ wb 207   ∧ wa 396  ∀wal 1545  ∃wex 1786

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1802  ax-4 1816  ax-5 1917  ax-6 1974  ax-7 2015  ax-8 2121  ax-9 2129  ax-10 2152  ax-11 2168  ax-12 2189  ax-13 2380  ax-ext 2712  ax-sep 5225  ax-nul 5235  ax-reg 9504

This theorem depends on definitions:  df-bi 208  df-an 397  df-or 854  df-tru 1550  df-ex 1787  df-nf 1791  df-cleq 2732  df-clel 2815  df-nfc 2889

This theorem is referenced by: (None)