Theorem axsepg4 35451

Description: A generalization of ax-sep 5251 that combines axsepg 5252 and axsepg2 35448 into a single theorem scheme. Unlike ax-sep 5251, this scheme lacks a distinct variable condition for 𝜑 and 𝑧 as well as for 𝑥 and 𝑧. Usage of this theorem is discouraged because it depends on ax-13 2406. (Contributed by BTernaryTau, 24-May-2026.) (New usage is discouraged.)

Assertion

Ref	Expression
axsepg4	⊢ ∃𝑦∀𝑥(𝑥 ∈ 𝑦 ↔ (𝑥 ∈ 𝑧 ∧ 𝜑))

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

Allowed substitution hints:   𝜑(𝑥,𝑧)

Proof of Theorem axsepg4

Dummy variable 𝑤 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		nfa1 2188	. . . 4 ⊢ Ⅎ𝑧∀𝑧∃𝑦∀𝑥(𝑥 ∈ 𝑦 ↔ (𝑥 ∈ 𝑤 ∧ 𝜑))
2	1	a1i 11	. . 3 ⊢ (¬ ∀𝑧 𝑧 = 𝑥 → Ⅎ𝑧∀𝑧∃𝑦∀𝑥(𝑥 ∈ 𝑦 ↔ (𝑥 ∈ 𝑤 ∧ 𝜑)))
3		nfvd 1938	. . 3 ⊢ (¬ ∀𝑧 𝑧 = 𝑥 → Ⅎ𝑤∃𝑦∀𝑥(𝑥 ∈ 𝑦 ↔ (𝑥 ∈ 𝑧 ∧ 𝜑)))
4		sp 2221	. . . 4 ⊢ (∀𝑧∃𝑦∀𝑥(𝑥 ∈ 𝑦 ↔ (𝑥 ∈ 𝑤 ∧ 𝜑)) → ∃𝑦∀𝑥(𝑥 ∈ 𝑦 ↔ (𝑥 ∈ 𝑤 ∧ 𝜑)))
5		dveeq2 2412	. . . . . 6 ⊢ (¬ ∀𝑥 𝑥 = 𝑧 → (𝑤 = 𝑧 → ∀𝑥 𝑤 = 𝑧))
6	5	naecoms 2463	. . . . 5 ⊢ (¬ ∀𝑧 𝑧 = 𝑥 → (𝑤 = 𝑧 → ∀𝑥 𝑤 = 𝑧))
7		elequ2 2160	. . . . . . . . . 10 ⊢ (𝑤 = 𝑧 → (𝑥 ∈ 𝑤 ↔ 𝑥 ∈ 𝑧))
8	7	anbi1d 642	. . . . . . . . 9 ⊢ (𝑤 = 𝑧 → ((𝑥 ∈ 𝑤 ∧ 𝜑) ↔ (𝑥 ∈ 𝑧 ∧ 𝜑)))
9	8	bibi2d 345	. . . . . . . 8 ⊢ (𝑤 = 𝑧 → ((𝑥 ∈ 𝑦 ↔ (𝑥 ∈ 𝑤 ∧ 𝜑)) ↔ (𝑥 ∈ 𝑦 ↔ (𝑥 ∈ 𝑧 ∧ 𝜑))))
10	9	biimpd 232	. . . . . . 7 ⊢ (𝑤 = 𝑧 → ((𝑥 ∈ 𝑦 ↔ (𝑥 ∈ 𝑤 ∧ 𝜑)) → (𝑥 ∈ 𝑦 ↔ (𝑥 ∈ 𝑧 ∧ 𝜑))))
11	10	al2imi 1838	. . . . . 6 ⊢ (∀𝑥 𝑤 = 𝑧 → (∀𝑥(𝑥 ∈ 𝑦 ↔ (𝑥 ∈ 𝑤 ∧ 𝜑)) → ∀𝑥(𝑥 ∈ 𝑦 ↔ (𝑥 ∈ 𝑧 ∧ 𝜑))))
12	11	eximdv 1940	. . . . 5 ⊢ (∀𝑥 𝑤 = 𝑧 → (∃𝑦∀𝑥(𝑥 ∈ 𝑦 ↔ (𝑥 ∈ 𝑤 ∧ 𝜑)) → ∃𝑦∀𝑥(𝑥 ∈ 𝑦 ↔ (𝑥 ∈ 𝑧 ∧ 𝜑))))
13	6, 12	syl6 36	. . . 4 ⊢ (¬ ∀𝑧 𝑧 = 𝑥 → (𝑤 = 𝑧 → (∃𝑦∀𝑥(𝑥 ∈ 𝑦 ↔ (𝑥 ∈ 𝑤 ∧ 𝜑)) → ∃𝑦∀𝑥(𝑥 ∈ 𝑦 ↔ (𝑥 ∈ 𝑧 ∧ 𝜑)))))
14	4, 13	syl7 75	. . 3 ⊢ (¬ ∀𝑧 𝑧 = 𝑥 → (𝑤 = 𝑧 → (∀𝑧∃𝑦∀𝑥(𝑥 ∈ 𝑦 ↔ (𝑥 ∈ 𝑤 ∧ 𝜑)) → ∃𝑦∀𝑥(𝑥 ∈ 𝑦 ↔ (𝑥 ∈ 𝑧 ∧ 𝜑)))))
15		elequ1 2152	. . . . . . . 8 ⊢ (𝑧 = 𝑥 → (𝑧 ∈ 𝑦 ↔ 𝑥 ∈ 𝑦))
16		elequ1 2152	. . . . . . . . 9 ⊢ (𝑧 = 𝑥 → (𝑧 ∈ 𝑧 ↔ 𝑥 ∈ 𝑧))
17	16	anbi1d 642	. . . . . . . 8 ⊢ (𝑧 = 𝑥 → ((𝑧 ∈ 𝑧 ∧ 𝜑) ↔ (𝑥 ∈ 𝑧 ∧ 𝜑)))
18	15, 17	bibi12d 348	. . . . . . 7 ⊢ (𝑧 = 𝑥 → ((𝑧 ∈ 𝑦 ↔ (𝑧 ∈ 𝑧 ∧ 𝜑)) ↔ (𝑥 ∈ 𝑦 ↔ (𝑥 ∈ 𝑧 ∧ 𝜑))))
19	18	biimpd 232	. . . . . 6 ⊢ (𝑧 = 𝑥 → ((𝑧 ∈ 𝑦 ↔ (𝑧 ∈ 𝑧 ∧ 𝜑)) → (𝑥 ∈ 𝑦 ↔ (𝑥 ∈ 𝑧 ∧ 𝜑))))
20	19	al2imi 1838	. . . . 5 ⊢ (∀𝑧 𝑧 = 𝑥 → (∀𝑧(𝑧 ∈ 𝑦 ↔ (𝑧 ∈ 𝑧 ∧ 𝜑)) → ∀𝑧(𝑥 ∈ 𝑦 ↔ (𝑥 ∈ 𝑧 ∧ 𝜑))))
21		axc11 2464	. . . . 5 ⊢ (∀𝑧 𝑧 = 𝑥 → (∀𝑧(𝑥 ∈ 𝑦 ↔ (𝑥 ∈ 𝑧 ∧ 𝜑)) → ∀𝑥(𝑥 ∈ 𝑦 ↔ (𝑥 ∈ 𝑧 ∧ 𝜑))))
22	20, 21	syld 48	. . . 4 ⊢ (∀𝑧 𝑧 = 𝑥 → (∀𝑧(𝑧 ∈ 𝑦 ↔ (𝑧 ∈ 𝑧 ∧ 𝜑)) → ∀𝑥(𝑥 ∈ 𝑦 ↔ (𝑥 ∈ 𝑧 ∧ 𝜑))))
23	22	eximdv 1940	. . 3 ⊢ (∀𝑧 𝑧 = 𝑥 → (∃𝑦∀𝑧(𝑧 ∈ 𝑦 ↔ (𝑧 ∈ 𝑧 ∧ 𝜑)) → ∃𝑦∀𝑥(𝑥 ∈ 𝑦 ↔ (𝑥 ∈ 𝑧 ∧ 𝜑))))
24		axsepg 5252	. . . 4 ⊢ ∃𝑦∀𝑥(𝑥 ∈ 𝑦 ↔ (𝑥 ∈ 𝑤 ∧ 𝜑))
25	24	gen2 1819	. . 3 ⊢ ∀𝑤∀𝑧∃𝑦∀𝑥(𝑥 ∈ 𝑦 ↔ (𝑥 ∈ 𝑤 ∧ 𝜑))
26		ax-nul 5261	. . . . 5 ⊢ ∃𝑦∀𝑧 ¬ 𝑧 ∈ 𝑦
27		elirrv 9547	. . . . . . . . 9 ⊢ ¬ 𝑧 ∈ 𝑧
28	27	intnanr 492	. . . . . . . 8 ⊢ ¬ (𝑧 ∈ 𝑧 ∧ 𝜑)
29	28	nbn 375	. . . . . . 7 ⊢ (¬ 𝑧 ∈ 𝑦 ↔ (𝑧 ∈ 𝑦 ↔ (𝑧 ∈ 𝑧 ∧ 𝜑)))
30	29	biimpi 219	. . . . . 6 ⊢ (¬ 𝑧 ∈ 𝑦 → (𝑧 ∈ 𝑦 ↔ (𝑧 ∈ 𝑧 ∧ 𝜑)))
31	30	alimi 1834	. . . . 5 ⊢ (∀𝑧 ¬ 𝑧 ∈ 𝑦 → ∀𝑧(𝑧 ∈ 𝑦 ↔ (𝑧 ∈ 𝑧 ∧ 𝜑)))
32	26, 31	eximii 1860	. . . 4 ⊢ ∃𝑦∀𝑧(𝑧 ∈ 𝑦 ↔ (𝑧 ∈ 𝑧 ∧ 𝜑))
33	32	ax-gen 1818	. . 3 ⊢ ∀𝑧∃𝑦∀𝑧(𝑧 ∈ 𝑦 ↔ (𝑧 ∈ 𝑧 ∧ 𝜑))
34	2, 3, 14, 23, 25, 33	dvelimalcasei 35381	. 2 ⊢ ∀𝑧∃𝑦∀𝑥(𝑥 ∈ 𝑦 ↔ (𝑥 ∈ 𝑧 ∧ 𝜑))
35	34	spi 2222	1 ⊢ ∃𝑦∀𝑥(𝑥 ∈ 𝑦 ↔ (𝑥 ∈ 𝑧 ∧ 𝜑))

Syntax hints:  ¬ wn 3   → wi 4   ↔ wb 209   ∧ wa 400  ∀wal 1561  ∃wex 1802  Ⅎwnf 1806

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1818  ax-4 1832  ax-5 1933  ax-6 1990  ax-7 2031  ax-8 2147  ax-9 2155  ax-10 2178  ax-11 2194  ax-12 2215  ax-13 2406  ax-sep 5251  ax-nul 5261  ax-reg 9542

This theorem depends on definitions:  df-bi 210  df-an 401  df-or 861  df-tru 1566  df-ex 1803  df-nf 1807

This theorem is referenced by:  axsepg5  35452