Theorem zfcndrep 10370

Description: Axiom of Replacement ax-rep 5209, reproved from conditionless ZFC axioms. Usage of this theorem is discouraged because it depends on ax-13 2372. (Contributed by NM, 15-Aug-2003.) (Proof modification is discouraged.) (New usage is discouraged.)

Assertion

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

Distinct variable group:   𝑥,𝑦,𝑧,𝑤

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

Proof of Theorem zfcndrep

Step	Hyp	Ref	Expression
1		nfe1 2147	. . . . . 6 ⊢ Ⅎ𝑦∃𝑦∀𝑧(∀𝑦𝜑 → 𝑧 = 𝑦)
2		nfv 1917	. . . . . . . 8 ⊢ Ⅎ𝑦 𝑧 ∈ 𝑤
3		nfv 1917	. . . . . . . . . 10 ⊢ Ⅎ𝑦 𝑤 ∈ 𝑥
4		nfa1 2148	. . . . . . . . . 10 ⊢ Ⅎ𝑦∀𝑦∀𝑦𝜑
5	3, 4	nfan 1902	. . . . . . . . 9 ⊢ Ⅎ𝑦(𝑤 ∈ 𝑥 ∧ ∀𝑦∀𝑦𝜑)
6	5	nfex 2318	. . . . . . . 8 ⊢ Ⅎ𝑦∃𝑤(𝑤 ∈ 𝑥 ∧ ∀𝑦∀𝑦𝜑)
7	2, 6	nfbi 1906	. . . . . . 7 ⊢ Ⅎ𝑦(𝑧 ∈ 𝑤 ↔ ∃𝑤(𝑤 ∈ 𝑥 ∧ ∀𝑦∀𝑦𝜑))
8	7	nfal 2317	. . . . . 6 ⊢ Ⅎ𝑦∀𝑧(𝑧 ∈ 𝑤 ↔ ∃𝑤(𝑤 ∈ 𝑥 ∧ ∀𝑦∀𝑦𝜑))
9	1, 8	nfim 1899	. . . . 5 ⊢ Ⅎ𝑦(∃𝑦∀𝑧(∀𝑦𝜑 → 𝑧 = 𝑦) → ∀𝑧(𝑧 ∈ 𝑤 ↔ ∃𝑤(𝑤 ∈ 𝑥 ∧ ∀𝑦∀𝑦𝜑)))
10	9	nfex 2318	. . . 4 ⊢ Ⅎ𝑦∃𝑤(∃𝑦∀𝑧(∀𝑦𝜑 → 𝑧 = 𝑦) → ∀𝑧(𝑧 ∈ 𝑤 ↔ ∃𝑤(𝑤 ∈ 𝑥 ∧ ∀𝑦∀𝑦𝜑)))
11		elequ2 2121	. . . . . . . . . 10 ⊢ (𝑦 = 𝑥 → (𝑤 ∈ 𝑦 ↔ 𝑤 ∈ 𝑥))
12	11	anbi1d 630	. . . . . . . . 9 ⊢ (𝑦 = 𝑥 → ((𝑤 ∈ 𝑦 ∧ ∀𝑦∀𝑦𝜑) ↔ (𝑤 ∈ 𝑥 ∧ ∀𝑦∀𝑦𝜑)))
13	12	exbidv 1924	. . . . . . . 8 ⊢ (𝑦 = 𝑥 → (∃𝑤(𝑤 ∈ 𝑦 ∧ ∀𝑦∀𝑦𝜑) ↔ ∃𝑤(𝑤 ∈ 𝑥 ∧ ∀𝑦∀𝑦𝜑)))
14	13	bibi2d 343	. . . . . . 7 ⊢ (𝑦 = 𝑥 → ((𝑧 ∈ 𝑤 ↔ ∃𝑤(𝑤 ∈ 𝑦 ∧ ∀𝑦∀𝑦𝜑)) ↔ (𝑧 ∈ 𝑤 ↔ ∃𝑤(𝑤 ∈ 𝑥 ∧ ∀𝑦∀𝑦𝜑))))
15	14	albidv 1923	. . . . . 6 ⊢ (𝑦 = 𝑥 → (∀𝑧(𝑧 ∈ 𝑤 ↔ ∃𝑤(𝑤 ∈ 𝑦 ∧ ∀𝑦∀𝑦𝜑)) ↔ ∀𝑧(𝑧 ∈ 𝑤 ↔ ∃𝑤(𝑤 ∈ 𝑥 ∧ ∀𝑦∀𝑦𝜑))))
16	15	imbi2d 341	. . . . 5 ⊢ (𝑦 = 𝑥 → ((∃𝑦∀𝑧(∀𝑦𝜑 → 𝑧 = 𝑦) → ∀𝑧(𝑧 ∈ 𝑤 ↔ ∃𝑤(𝑤 ∈ 𝑦 ∧ ∀𝑦∀𝑦𝜑))) ↔ (∃𝑦∀𝑧(∀𝑦𝜑 → 𝑧 = 𝑦) → ∀𝑧(𝑧 ∈ 𝑤 ↔ ∃𝑤(𝑤 ∈ 𝑥 ∧ ∀𝑦∀𝑦𝜑)))))
17	16	exbidv 1924	. . . 4 ⊢ (𝑦 = 𝑥 → (∃𝑤(∃𝑦∀𝑧(∀𝑦𝜑 → 𝑧 = 𝑦) → ∀𝑧(𝑧 ∈ 𝑤 ↔ ∃𝑤(𝑤 ∈ 𝑦 ∧ ∀𝑦∀𝑦𝜑))) ↔ ∃𝑤(∃𝑦∀𝑧(∀𝑦𝜑 → 𝑧 = 𝑦) → ∀𝑧(𝑧 ∈ 𝑤 ↔ ∃𝑤(𝑤 ∈ 𝑥 ∧ ∀𝑦∀𝑦𝜑)))))
18		axrepnd 10350	. . . . 5 ⊢ ∃𝑤(∃𝑦∀𝑧(∀𝑦𝜑 → 𝑧 = 𝑦) → ∀𝑧(∀𝑦 𝑧 ∈ 𝑤 ↔ ∃𝑤(∀𝑧 𝑤 ∈ 𝑦 ∧ ∀𝑦∀𝑦𝜑)))
19		19.3v 1985	. . . . . . . . 9 ⊢ (∀𝑦 𝑧 ∈ 𝑤 ↔ 𝑧 ∈ 𝑤)
20		19.3v 1985	. . . . . . . . . . 11 ⊢ (∀𝑧 𝑤 ∈ 𝑦 ↔ 𝑤 ∈ 𝑦)
21	20	anbi1i 624	. . . . . . . . . 10 ⊢ ((∀𝑧 𝑤 ∈ 𝑦 ∧ ∀𝑦∀𝑦𝜑) ↔ (𝑤 ∈ 𝑦 ∧ ∀𝑦∀𝑦𝜑))
22	21	exbii 1850	. . . . . . . . 9 ⊢ (∃𝑤(∀𝑧 𝑤 ∈ 𝑦 ∧ ∀𝑦∀𝑦𝜑) ↔ ∃𝑤(𝑤 ∈ 𝑦 ∧ ∀𝑦∀𝑦𝜑))
23	19, 22	bibi12i 340	. . . . . . . 8 ⊢ ((∀𝑦 𝑧 ∈ 𝑤 ↔ ∃𝑤(∀𝑧 𝑤 ∈ 𝑦 ∧ ∀𝑦∀𝑦𝜑)) ↔ (𝑧 ∈ 𝑤 ↔ ∃𝑤(𝑤 ∈ 𝑦 ∧ ∀𝑦∀𝑦𝜑)))
24	23	albii 1822	. . . . . . 7 ⊢ (∀𝑧(∀𝑦 𝑧 ∈ 𝑤 ↔ ∃𝑤(∀𝑧 𝑤 ∈ 𝑦 ∧ ∀𝑦∀𝑦𝜑)) ↔ ∀𝑧(𝑧 ∈ 𝑤 ↔ ∃𝑤(𝑤 ∈ 𝑦 ∧ ∀𝑦∀𝑦𝜑)))
25	24	imbi2i 336	. . . . . 6 ⊢ ((∃𝑦∀𝑧(∀𝑦𝜑 → 𝑧 = 𝑦) → ∀𝑧(∀𝑦 𝑧 ∈ 𝑤 ↔ ∃𝑤(∀𝑧 𝑤 ∈ 𝑦 ∧ ∀𝑦∀𝑦𝜑))) ↔ (∃𝑦∀𝑧(∀𝑦𝜑 → 𝑧 = 𝑦) → ∀𝑧(𝑧 ∈ 𝑤 ↔ ∃𝑤(𝑤 ∈ 𝑦 ∧ ∀𝑦∀𝑦𝜑))))
26	25	exbii 1850	. . . . 5 ⊢ (∃𝑤(∃𝑦∀𝑧(∀𝑦𝜑 → 𝑧 = 𝑦) → ∀𝑧(∀𝑦 𝑧 ∈ 𝑤 ↔ ∃𝑤(∀𝑧 𝑤 ∈ 𝑦 ∧ ∀𝑦∀𝑦𝜑))) ↔ ∃𝑤(∃𝑦∀𝑧(∀𝑦𝜑 → 𝑧 = 𝑦) → ∀𝑧(𝑧 ∈ 𝑤 ↔ ∃𝑤(𝑤 ∈ 𝑦 ∧ ∀𝑦∀𝑦𝜑))))
27	18, 26	mpbi 229	. . . 4 ⊢ ∃𝑤(∃𝑦∀𝑧(∀𝑦𝜑 → 𝑧 = 𝑦) → ∀𝑧(𝑧 ∈ 𝑤 ↔ ∃𝑤(𝑤 ∈ 𝑦 ∧ ∀𝑦∀𝑦𝜑)))
28	10, 17, 27	chvar 2395	. . 3 ⊢ ∃𝑤(∃𝑦∀𝑧(∀𝑦𝜑 → 𝑧 = 𝑦) → ∀𝑧(𝑧 ∈ 𝑤 ↔ ∃𝑤(𝑤 ∈ 𝑥 ∧ ∀𝑦∀𝑦𝜑)))
29	28	19.35i 1881	. 2 ⊢ (∀𝑤∃𝑦∀𝑧(∀𝑦𝜑 → 𝑧 = 𝑦) → ∃𝑤∀𝑧(𝑧 ∈ 𝑤 ↔ ∃𝑤(𝑤 ∈ 𝑥 ∧ ∀𝑦∀𝑦𝜑)))
30		nfv 1917	. . . . 5 ⊢ Ⅎ𝑤 𝑧 ∈ 𝑦
31		nfe1 2147	. . . . 5 ⊢ Ⅎ𝑤∃𝑤(𝑤 ∈ 𝑥 ∧ ∀𝑦𝜑)
32	30, 31	nfbi 1906	. . . 4 ⊢ Ⅎ𝑤(𝑧 ∈ 𝑦 ↔ ∃𝑤(𝑤 ∈ 𝑥 ∧ ∀𝑦𝜑))
33	32	nfal 2317	. . 3 ⊢ Ⅎ𝑤∀𝑧(𝑧 ∈ 𝑦 ↔ ∃𝑤(𝑤 ∈ 𝑥 ∧ ∀𝑦𝜑))
34		elequ2 2121	. . . . 5 ⊢ (𝑤 = 𝑦 → (𝑧 ∈ 𝑤 ↔ 𝑧 ∈ 𝑦))
35		nfa1 2148	. . . . . . . . 9 ⊢ Ⅎ𝑦∀𝑦𝜑
36	35	19.3 2195	. . . . . . . 8 ⊢ (∀𝑦∀𝑦𝜑 ↔ ∀𝑦𝜑)
37	36	anbi2i 623	. . . . . . 7 ⊢ ((𝑤 ∈ 𝑥 ∧ ∀𝑦∀𝑦𝜑) ↔ (𝑤 ∈ 𝑥 ∧ ∀𝑦𝜑))
38	37	exbii 1850	. . . . . 6 ⊢ (∃𝑤(𝑤 ∈ 𝑥 ∧ ∀𝑦∀𝑦𝜑) ↔ ∃𝑤(𝑤 ∈ 𝑥 ∧ ∀𝑦𝜑))
39	38	a1i 11	. . . . 5 ⊢ (𝑤 = 𝑦 → (∃𝑤(𝑤 ∈ 𝑥 ∧ ∀𝑦∀𝑦𝜑) ↔ ∃𝑤(𝑤 ∈ 𝑥 ∧ ∀𝑦𝜑)))
40	34, 39	bibi12d 346	. . . 4 ⊢ (𝑤 = 𝑦 → ((𝑧 ∈ 𝑤 ↔ ∃𝑤(𝑤 ∈ 𝑥 ∧ ∀𝑦∀𝑦𝜑)) ↔ (𝑧 ∈ 𝑦 ↔ ∃𝑤(𝑤 ∈ 𝑥 ∧ ∀𝑦𝜑))))
41	40	albidv 1923	. . 3 ⊢ (𝑤 = 𝑦 → (∀𝑧(𝑧 ∈ 𝑤 ↔ ∃𝑤(𝑤 ∈ 𝑥 ∧ ∀𝑦∀𝑦𝜑)) ↔ ∀𝑧(𝑧 ∈ 𝑦 ↔ ∃𝑤(𝑤 ∈ 𝑥 ∧ ∀𝑦𝜑))))
42	8, 33, 41	cbvexv1 2339	. 2 ⊢ (∃𝑤∀𝑧(𝑧 ∈ 𝑤 ↔ ∃𝑤(𝑤 ∈ 𝑥 ∧ ∀𝑦∀𝑦𝜑)) ↔ ∃𝑦∀𝑧(𝑧 ∈ 𝑦 ↔ ∃𝑤(𝑤 ∈ 𝑥 ∧ ∀𝑦𝜑)))
43	29, 42	sylib 217	1 ⊢ (∀𝑤∃𝑦∀𝑧(∀𝑦𝜑 → 𝑧 = 𝑦) → ∃𝑦∀𝑧(𝑧 ∈ 𝑦 ↔ ∃𝑤(𝑤 ∈ 𝑥 ∧ ∀𝑦𝜑)))

Syntax hints:   → wi 4   ↔ wb 205   ∧ wa 396  ∀wal 1537   = wceq 1539  ∃wex 1782   ∈ wcel 2106

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-10 2137  ax-11 2154  ax-12 2171  ax-13 2372  ax-ext 2709  ax-rep 5209  ax-sep 5223  ax-nul 5230  ax-pr 5352  ax-reg 9351

This theorem depends on definitions:  df-bi 206  df-an 397  df-or 845  df-tru 1542  df-fal 1552  df-ex 1783  df-nf 1787  df-sb 2068  df-clab 2716  df-cleq 2730  df-clel 2816  df-nfc 2889  df-ral 3069  df-rex 3070  df-v 3434  df-dif 3890  df-un 3892  df-nul 4257  df-sn 4562  df-pr 4564

This theorem is referenced by: (None)