Theorem axrep4OLD 5286

Description: Obsolete version of axrep4 5285 as of 18-Sep-2025. (Contributed by NM, 14-Aug-1994.) (Proof modification is discouraged.) (New usage is discouraged.)

Hypothesis

Ref	Expression
axrep4OLD.1	⊢ Ⅎ𝑧𝜑

Assertion

Ref	Expression
axrep4OLD	⊢ (∀𝑥∃𝑧∀𝑦(𝜑 → 𝑦 = 𝑧) → ∃𝑧∀𝑦(𝑦 ∈ 𝑧 ↔ ∃𝑥(𝑥 ∈ 𝑤 ∧ 𝜑)))

Distinct variable group:   𝑥,𝑦,𝑧,𝑤

Allowed substitution hints:   𝜑(𝑥,𝑦,𝑧,𝑤)

Proof of Theorem axrep4OLD

Step	Hyp	Ref	Expression
1		axrep3 5283	. . 3 ⊢ ∃𝑥(∃𝑧∀𝑦(𝜑 → 𝑦 = 𝑧) → ∀𝑦(𝑦 ∈ 𝑥 ↔ ∃𝑥(𝑥 ∈ 𝑤 ∧ ∀𝑧𝜑)))
2	1	19.35i 1878	. 2 ⊢ (∀𝑥∃𝑧∀𝑦(𝜑 → 𝑦 = 𝑧) → ∃𝑥∀𝑦(𝑦 ∈ 𝑥 ↔ ∃𝑥(𝑥 ∈ 𝑤 ∧ ∀𝑧𝜑)))
3		nfv 1914	. . . . 5 ⊢ Ⅎ𝑧 𝑦 ∈ 𝑥
4		nfv 1914	. . . . . . 7 ⊢ Ⅎ𝑧 𝑥 ∈ 𝑤
5		nfa1 2151	. . . . . . 7 ⊢ Ⅎ𝑧∀𝑧𝜑
6	4, 5	nfan 1899	. . . . . 6 ⊢ Ⅎ𝑧(𝑥 ∈ 𝑤 ∧ ∀𝑧𝜑)
7	6	nfex 2324	. . . . 5 ⊢ Ⅎ𝑧∃𝑥(𝑥 ∈ 𝑤 ∧ ∀𝑧𝜑)
8	3, 7	nfbi 1903	. . . 4 ⊢ Ⅎ𝑧(𝑦 ∈ 𝑥 ↔ ∃𝑥(𝑥 ∈ 𝑤 ∧ ∀𝑧𝜑))
9	8	nfal 2323	. . 3 ⊢ Ⅎ𝑧∀𝑦(𝑦 ∈ 𝑥 ↔ ∃𝑥(𝑥 ∈ 𝑤 ∧ ∀𝑧𝜑))
10		nfv 1914	. . . . 5 ⊢ Ⅎ𝑥 𝑦 ∈ 𝑧
11		nfe1 2150	. . . . 5 ⊢ Ⅎ𝑥∃𝑥(𝑥 ∈ 𝑤 ∧ 𝜑)
12	10, 11	nfbi 1903	. . . 4 ⊢ Ⅎ𝑥(𝑦 ∈ 𝑧 ↔ ∃𝑥(𝑥 ∈ 𝑤 ∧ 𝜑))
13	12	nfal 2323	. . 3 ⊢ Ⅎ𝑥∀𝑦(𝑦 ∈ 𝑧 ↔ ∃𝑥(𝑥 ∈ 𝑤 ∧ 𝜑))
14		elequ2 2123	. . . . 5 ⊢ (𝑥 = 𝑧 → (𝑦 ∈ 𝑥 ↔ 𝑦 ∈ 𝑧))
15		axrep4OLD.1	. . . . . . . . 9 ⊢ Ⅎ𝑧𝜑
16	15	19.3 2202	. . . . . . . 8 ⊢ (∀𝑧𝜑 ↔ 𝜑)
17	16	anbi2i 623	. . . . . . 7 ⊢ ((𝑥 ∈ 𝑤 ∧ ∀𝑧𝜑) ↔ (𝑥 ∈ 𝑤 ∧ 𝜑))
18	17	exbii 1848	. . . . . 6 ⊢ (∃𝑥(𝑥 ∈ 𝑤 ∧ ∀𝑧𝜑) ↔ ∃𝑥(𝑥 ∈ 𝑤 ∧ 𝜑))
19	18	a1i 11	. . . . 5 ⊢ (𝑥 = 𝑧 → (∃𝑥(𝑥 ∈ 𝑤 ∧ ∀𝑧𝜑) ↔ ∃𝑥(𝑥 ∈ 𝑤 ∧ 𝜑)))
20	14, 19	bibi12d 345	. . . 4 ⊢ (𝑥 = 𝑧 → ((𝑦 ∈ 𝑥 ↔ ∃𝑥(𝑥 ∈ 𝑤 ∧ ∀𝑧𝜑)) ↔ (𝑦 ∈ 𝑧 ↔ ∃𝑥(𝑥 ∈ 𝑤 ∧ 𝜑))))
21	20	albidv 1920	. . 3 ⊢ (𝑥 = 𝑧 → (∀𝑦(𝑦 ∈ 𝑥 ↔ ∃𝑥(𝑥 ∈ 𝑤 ∧ ∀𝑧𝜑)) ↔ ∀𝑦(𝑦 ∈ 𝑧 ↔ ∃𝑥(𝑥 ∈ 𝑤 ∧ 𝜑))))
22	9, 13, 21	cbvexv1 2344	. 2 ⊢ (∃𝑥∀𝑦(𝑦 ∈ 𝑥 ↔ ∃𝑥(𝑥 ∈ 𝑤 ∧ ∀𝑧𝜑)) ↔ ∃𝑧∀𝑦(𝑦 ∈ 𝑧 ↔ ∃𝑥(𝑥 ∈ 𝑤 ∧ 𝜑)))
23	2, 22	sylib 218	1 ⊢ (∀𝑥∃𝑧∀𝑦(𝜑 → 𝑦 = 𝑧) → ∃𝑧∀𝑦(𝑦 ∈ 𝑧 ↔ ∃𝑥(𝑥 ∈ 𝑤 ∧ 𝜑)))

Syntax hints:   → wi 4   ↔ wb 206   ∧ wa 395  ∀wal 1538  ∃wex 1779  Ⅎwnf 1783

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2007  ax-9 2118  ax-10 2141  ax-11 2157  ax-12 2177  ax-rep 5279

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-tru 1543  df-ex 1780  df-nf 1784

This theorem is referenced by: (None)