Theorem axsepg5 35581

Description: A generalization of ax-sep 5262 that combines axsepg 5263, axsepg2 35577, and axsepg3 35578 into a single theorem scheme. Unlike ax-sep 5262, this scheme lacks a distinct variable condition for 𝜑 and 𝑧, for 𝑥 and 𝑧, and for 𝑦 and 𝑧. Usage of this theorem is discouraged because it depends on ax-13 2407. (Contributed by BTernaryTau, 24-May-2026.) (New usage is discouraged.)

Assertion

Ref	Expression
axsepg5	⊢ ∃𝑦∀𝑥(𝑥 ∈ 𝑦 ↔ (𝑥 ∈ 𝑧 ∧ 𝜑))

Distinct variable groups:   𝜑,𝑦   𝑥,𝑦

Allowed substitution hints:   𝜑(𝑥, 𝑧)

Proof of Theorem axsepg5

Dummy variable 𝑤 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		nfnae 2469	. . 3 ⊢ Ⅎ𝑥 ¬ ∀𝑦 𝑦 = 𝑧
2		nfvd 1948	. . . 4 ⊢ (¬ ∀𝑦 𝑦 = 𝑧 → Ⅎ𝑦 𝑥 ∈ 𝑤)
3		nfcvf 2954	. . . . . 6 ⊢ (¬ ∀𝑦 𝑦 = 𝑧 → Ⅎ𝑦𝑧)
4	3	nfcrd 2922	. . . . 5 ⊢ (¬ ∀𝑦 𝑦 = 𝑧 → Ⅎ𝑦 𝑥 ∈ 𝑧)
5		nfvd 1948	. . . . 5 ⊢ (¬ ∀𝑦 𝑦 = 𝑧 → Ⅎ𝑦𝜑)
6	4, 5	nfand 1930	. . . 4 ⊢ (¬ ∀𝑦 𝑦 = 𝑧 → Ⅎ𝑦(𝑥 ∈ 𝑧 ∧ 𝜑))
7	2, 6	nfbid 1935	. . 3 ⊢ (¬ ∀𝑦 𝑦 = 𝑧 → Ⅎ𝑦(𝑥 ∈ 𝑤 ↔ (𝑥 ∈ 𝑧 ∧ 𝜑)))
8	1, 7	nfald 2364	. 2 ⊢ (¬ ∀𝑦 𝑦 = 𝑧 → Ⅎ𝑦∀𝑥(𝑥 ∈ 𝑤 ↔ (𝑥 ∈ 𝑧 ∧ 𝜑)))
9		nfvd 1948	. 2 ⊢ (¬ ∀𝑦 𝑦 = 𝑧 → Ⅎ𝑤∀𝑥(𝑥 ∈ 𝑦 ↔ (𝑥 ∈ 𝑧 ∧ 𝜑)))
10		elequ2 2161	. . . . . 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	albidv 1953	. . . 4 ⊢ (𝑤 = 𝑦 → (∀𝑥(𝑥 ∈ 𝑤 ↔ (𝑥 ∈ 𝑧 ∧ 𝜑)) ↔ ∀𝑥(𝑥 ∈ 𝑦 ↔ (𝑥 ∈ 𝑧 ∧ 𝜑))))
13	12	biimpd 232	. . 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		nfae 2468	. . 3 ⊢ Ⅎ𝑥∀𝑦 𝑦 = 𝑧
16		elequ2 2161	. . . . . . 7 ⊢ (𝑦 = 𝑧 → (𝑥 ∈ 𝑦 ↔ 𝑥 ∈ 𝑧))
17	16	anbi1d 643	. . . . . 6 ⊢ (𝑦 = 𝑧 → ((𝑥 ∈ 𝑦 ∧ 𝜑) ↔ (𝑥 ∈ 𝑧 ∧ 𝜑)))
18	17	bibi2d 345	. . . . 5 ⊢ (𝑦 = 𝑧 → ((𝑥 ∈ 𝑦 ↔ (𝑥 ∈ 𝑦 ∧ 𝜑)) ↔ (𝑥 ∈ 𝑦 ↔ (𝑥 ∈ 𝑧 ∧ 𝜑))))
19	18	biimpd 232	. . . 4 ⊢ (𝑦 = 𝑧 → ((𝑥 ∈ 𝑦 ↔ (𝑥 ∈ 𝑦 ∧ 𝜑)) → (𝑥 ∈ 𝑦 ↔ (𝑥 ∈ 𝑧 ∧ 𝜑))))
20	19	sps 2224	. . 3 ⊢ (∀𝑦 𝑦 = 𝑧 → ((𝑥 ∈ 𝑦 ↔ (𝑥 ∈ 𝑦 ∧ 𝜑)) → (𝑥 ∈ 𝑦 ↔ (𝑥 ∈ 𝑧 ∧ 𝜑))))
21	15, 20	alimd 2251	. 2 ⊢ (∀𝑦 𝑦 = 𝑧 → (∀𝑥(𝑥 ∈ 𝑦 ↔ (𝑥 ∈ 𝑦 ∧ 𝜑)) → ∀𝑥(𝑥 ∈ 𝑦 ↔ (𝑥 ∈ 𝑧 ∧ 𝜑))))
22		axsepg4 35580	. 2 ⊢ ∃𝑤∀𝑥(𝑥 ∈ 𝑤 ↔ (𝑥 ∈ 𝑧 ∧ 𝜑))
23		axsepg3 35578	. 2 ⊢ ∃𝑦∀𝑥(𝑥 ∈ 𝑦 ↔ (𝑥 ∈ 𝑦 ∧ 𝜑))
24	8, 9, 14, 21, 22, 23	dvelimexcasei 35498	1 ⊢ ∃𝑦∀𝑥(𝑥 ∈ 𝑦 ↔ (𝑥 ∈ 𝑧 ∧ 𝜑))

This proof depends on syntax axioms:  ¬ wn 3   → wi 4   ↔ wb 209   ∧ wa 401  ∀wal 1568  ∃wex 1812

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1828  ax-4 1842  ax-5 1943  ax-6 2000  ax-7 2041  ax-8 2148  ax-9 2156  ax-10 2179  ax-11 2195  ax-12 2216  ax-13 2407  ax-sep 5262  ax-nul 5274  ax-reg 9564

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-tru 1573  df-ex 1813  df-nf 1817  df-clel 2841  df-nfc 2915

This theorem is used by: (None)