Theorem axsepg3ALT 35513

Description: Alternate proof of axsepg3 35512, derived directly from ax-sep 5262 with no additional set theory axioms. (Contributed by BTernaryTau, 3-Aug-2025.) (Proof modification is discouraged.) (New usage is discouraged.)

Assertion

Ref	Expression
axsepg3ALT	⊢ ∃𝑦∀𝑥(𝑥 ∈ 𝑦 ↔ (𝑥 ∈ 𝑧 ∧ 𝜑))

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

Allowed substitution hint:   𝜑(𝑥)

Proof of Theorem axsepg3ALT

Dummy variables 𝑤 𝑣 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		nfv 1942	. . 3 ⊢ Ⅎ𝑥 ¬ ∀𝑦 𝑦 = 𝑧
2		nfvd 1943	. . . 4 ⊢ (¬ ∀𝑦 𝑦 = 𝑧 → Ⅎ𝑦 𝑥 ∈ 𝑤)
3		nfcvf 2958	. . . . . 6 ⊢ (¬ ∀𝑦 𝑦 = 𝑧 → Ⅎ𝑦𝑧)
4	3	nfcrd 2926	. . . . 5 ⊢ (¬ ∀𝑦 𝑦 = 𝑧 → Ⅎ𝑦 𝑥 ∈ 𝑧)
5		nfvd 1943	. . . . 5 ⊢ (¬ ∀𝑦 𝑦 = 𝑧 → Ⅎ𝑦𝜑)
6	4, 5	nfand 1925	. . . 4 ⊢ (¬ ∀𝑦 𝑦 = 𝑧 → Ⅎ𝑦(𝑥 ∈ 𝑧 ∧ 𝜑))
7	2, 6	nfbid 1930	. . 3 ⊢ (¬ ∀𝑦 𝑦 = 𝑧 → Ⅎ𝑦(𝑥 ∈ 𝑤 ↔ (𝑥 ∈ 𝑧 ∧ 𝜑)))
8	1, 7	nfald 2368	. 2 ⊢ (¬ ∀𝑦 𝑦 = 𝑧 → Ⅎ𝑦∀𝑥(𝑥 ∈ 𝑤 ↔ (𝑥 ∈ 𝑧 ∧ 𝜑)))
9		nfvd 1943	. 2 ⊢ (¬ ∀𝑦 𝑦 = 𝑧 → Ⅎ𝑤∀𝑥(𝑥 ∈ 𝑦 ↔ (𝑥 ∈ 𝑧 ∧ 𝜑)))
10		elequ2 2165	. . . . . 6 ⊢ (𝑤 = 𝑦 → (𝑥 ∈ 𝑤 ↔ 𝑥 ∈ 𝑦))
11	10	bibi1d 346	. . . . 5 ⊢ (𝑤 = 𝑦 → ((𝑥 ∈ 𝑤 ↔ (𝑥 ∈ 𝑧 ∧ 𝜑)) ↔ (𝑥 ∈ 𝑦 ↔ (𝑥 ∈ 𝑧 ∧ 𝜑))))
12	11	biimpd 232	. . . 4 ⊢ (𝑤 = 𝑦 → ((𝑥 ∈ 𝑤 ↔ (𝑥 ∈ 𝑧 ∧ 𝜑)) → (𝑥 ∈ 𝑦 ↔ (𝑥 ∈ 𝑧 ∧ 𝜑))))
13	12	alimdv 1944	. . 3 ⊢ (𝑤 = 𝑦 → (∀𝑥(𝑥 ∈ 𝑤 ↔ (𝑥 ∈ 𝑧 ∧ 𝜑)) → ∀𝑥(𝑥 ∈ 𝑦 ↔ (𝑥 ∈ 𝑧 ∧ 𝜑))))
14	13	a1i 11	. 2 ⊢ (¬ ∀𝑦 𝑦 = 𝑧 → (𝑤 = 𝑦 → (∀𝑥(𝑥 ∈ 𝑤 ↔ (𝑥 ∈ 𝑧 ∧ 𝜑)) → ∀𝑥(𝑥 ∈ 𝑦 ↔ (𝑥 ∈ 𝑧 ∧ 𝜑)))))
15		elequ2 2165	. . . . . . 7 ⊢ (𝑦 = 𝑧 → (𝑥 ∈ 𝑦 ↔ 𝑥 ∈ 𝑧))
16	15	anbi1d 642	. . . . . 6 ⊢ (𝑦 = 𝑧 → ((𝑥 ∈ 𝑦 ∧ 𝜑) ↔ (𝑥 ∈ 𝑧 ∧ 𝜑)))
17	16	bibi2d 345	. . . . 5 ⊢ (𝑦 = 𝑧 → ((𝑥 ∈ 𝑦 ↔ (𝑥 ∈ 𝑦 ∧ 𝜑)) ↔ (𝑥 ∈ 𝑦 ↔ (𝑥 ∈ 𝑧 ∧ 𝜑))))
18	17	biimpd 232	. . . 4 ⊢ (𝑦 = 𝑧 → ((𝑥 ∈ 𝑦 ↔ (𝑥 ∈ 𝑦 ∧ 𝜑)) → (𝑥 ∈ 𝑦 ↔ (𝑥 ∈ 𝑧 ∧ 𝜑))))
19	18	alimdv 1944	. . 3 ⊢ (𝑦 = 𝑧 → (∀𝑥(𝑥 ∈ 𝑦 ↔ (𝑥 ∈ 𝑦 ∧ 𝜑)) → ∀𝑥(𝑥 ∈ 𝑦 ↔ (𝑥 ∈ 𝑧 ∧ 𝜑))))
20	19	sps 2228	. 2 ⊢ (∀𝑦 𝑦 = 𝑧 → (∀𝑥(𝑥 ∈ 𝑦 ↔ (𝑥 ∈ 𝑦 ∧ 𝜑)) → ∀𝑥(𝑥 ∈ 𝑦 ↔ (𝑥 ∈ 𝑧 ∧ 𝜑))))
21		ax-sep 5262	. 2 ⊢ ∃𝑤∀𝑥(𝑥 ∈ 𝑤 ↔ (𝑥 ∈ 𝑧 ∧ 𝜑))
22		ax-sep 5262	. . 3 ⊢ ∃𝑦∀𝑥(𝑥 ∈ 𝑦 ↔ (𝑥 ∈ 𝑣 ∧ ⊥))
23		fal 1582	. . . . . . 7 ⊢ ¬ ⊥
24	23	intnan 491	. . . . . 6 ⊢ ¬ (𝑥 ∈ 𝑣 ∧ ⊥)
25		biimp 218	. . . . . 6 ⊢ ((𝑥 ∈ 𝑦 ↔ (𝑥 ∈ 𝑣 ∧ ⊥)) → (𝑥 ∈ 𝑦 → (𝑥 ∈ 𝑣 ∧ ⊥)))
26	24, 25	mtoi 202	. . . . 5 ⊢ ((𝑥 ∈ 𝑦 ↔ (𝑥 ∈ 𝑣 ∧ ⊥)) → ¬ 𝑥 ∈ 𝑦)
27	26	bianfd 543	. . . 4 ⊢ ((𝑥 ∈ 𝑦 ↔ (𝑥 ∈ 𝑣 ∧ ⊥)) → (𝑥 ∈ 𝑦 ↔ (𝑥 ∈ 𝑦 ∧ 𝜑)))
28	27	alimi 1839	. . 3 ⊢ (∀𝑥(𝑥 ∈ 𝑦 ↔ (𝑥 ∈ 𝑣 ∧ ⊥)) → ∀𝑥(𝑥 ∈ 𝑦 ↔ (𝑥 ∈ 𝑦 ∧ 𝜑)))
29	22, 28	eximii 1865	. 2 ⊢ ∃𝑦∀𝑥(𝑥 ∈ 𝑦 ↔ (𝑥 ∈ 𝑦 ∧ 𝜑))
30	8, 9, 14, 20, 21, 29	dvelimexcasei 35436	1 ⊢ ∃𝑦∀𝑥(𝑥 ∈ 𝑦 ↔ (𝑥 ∈ 𝑧 ∧ 𝜑))

Syntax hints:  ¬ wn 3   → wi 4   ↔ wb 209   ∧ wa 400  ∀wal 1566  ⊥wfal 1580  ∃wex 1807

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1823  ax-4 1837  ax-5 1938  ax-6 1995  ax-7 2036  ax-8 2152  ax-9 2160  ax-10 2183  ax-11 2199  ax-12 2220  ax-13 2411  ax-sep 5262

This theorem depends on definitions:  df-bi 210  df-an 401  df-or 861  df-tru 1571  df-fal 1581  df-ex 1808  df-nf 1812  df-clel 2845  df-nfc 2919

This theorem is referenced by: (None)