Theorem strcollnft 13866

Description: Closed form of strcollnf 13867. (Contributed by BJ, 21-Oct-2019.)

Assertion

Ref	Expression
strcollnft	⊢ (∀𝑥∀𝑦Ⅎ𝑏𝜑 → (∀𝑥 ∈ 𝑎 ∃𝑦𝜑 → ∃𝑏(∀𝑥 ∈ 𝑎 ∃𝑦 ∈ 𝑏 𝜑 ∧ ∀𝑦 ∈ 𝑏 ∃𝑥 ∈ 𝑎 𝜑)))

Distinct variable group:   𝑎,𝑏,𝑥,𝑦

Allowed substitution hints:   𝜑(𝑥,𝑦,𝑎,𝑏)

Proof of Theorem strcollnft

Dummy variable 𝑧 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		strcoll2 13865	. 2 ⊢ (∀𝑥 ∈ 𝑎 ∃𝑦𝜑 → ∃𝑧(∀𝑥 ∈ 𝑎 ∃𝑦 ∈ 𝑧 𝜑 ∧ ∀𝑦 ∈ 𝑧 ∃𝑥 ∈ 𝑎 𝜑))
2		nfnf1 1532	. . . . 5 ⊢ Ⅎ𝑏Ⅎ𝑏𝜑
3	2	nfal 1564	. . . 4 ⊢ Ⅎ𝑏∀𝑦Ⅎ𝑏𝜑
4	3	nfal 1564	. . 3 ⊢ Ⅎ𝑏∀𝑥∀𝑦Ⅎ𝑏𝜑
5		nfa1 1529	. . . . 5 ⊢ Ⅎ𝑥∀𝑥∀𝑦Ⅎ𝑏𝜑
6		nfcvd 2309	. . . . 5 ⊢ (∀𝑥∀𝑦Ⅎ𝑏𝜑 → Ⅎ𝑏𝑎)
7		nfa1 1529	. . . . . . 7 ⊢ Ⅎ𝑦∀𝑦Ⅎ𝑏𝜑
8	7	nfal 1564	. . . . . 6 ⊢ Ⅎ𝑦∀𝑥∀𝑦Ⅎ𝑏𝜑
9		nfcvd 2309	. . . . . 6 ⊢ (∀𝑥∀𝑦Ⅎ𝑏𝜑 → Ⅎ𝑏𝑧)
10		sp 1499	. . . . . . 7 ⊢ (∀𝑦Ⅎ𝑏𝜑 → Ⅎ𝑏𝜑)
11	10	sps 1525	. . . . . 6 ⊢ (∀𝑥∀𝑦Ⅎ𝑏𝜑 → Ⅎ𝑏𝜑)
12	8, 9, 11	nfrexdxy 2500	. . . . 5 ⊢ (∀𝑥∀𝑦Ⅎ𝑏𝜑 → Ⅎ𝑏∃𝑦 ∈ 𝑧 𝜑)
13	5, 6, 12	nfraldxy 2499	. . . 4 ⊢ (∀𝑥∀𝑦Ⅎ𝑏𝜑 → Ⅎ𝑏∀𝑥 ∈ 𝑎 ∃𝑦 ∈ 𝑧 𝜑)
14	5, 6, 11	nfrexdxy 2500	. . . . 5 ⊢ (∀𝑥∀𝑦Ⅎ𝑏𝜑 → Ⅎ𝑏∃𝑥 ∈ 𝑎 𝜑)
15	8, 9, 14	nfraldxy 2499	. . . 4 ⊢ (∀𝑥∀𝑦Ⅎ𝑏𝜑 → Ⅎ𝑏∀𝑦 ∈ 𝑧 ∃𝑥 ∈ 𝑎 𝜑)
16	13, 15	nfand 1556	. . 3 ⊢ (∀𝑥∀𝑦Ⅎ𝑏𝜑 → Ⅎ𝑏(∀𝑥 ∈ 𝑎 ∃𝑦 ∈ 𝑧 𝜑 ∧ ∀𝑦 ∈ 𝑧 ∃𝑥 ∈ 𝑎 𝜑))
17		nfv 1516	. . . . . . 7 ⊢ Ⅎ𝑥 𝑧 = 𝑏
18	5, 17	nfan 1553	. . . . . 6 ⊢ Ⅎ𝑥(∀𝑥∀𝑦Ⅎ𝑏𝜑 ∧ 𝑧 = 𝑏)
19		rexeq 2662	. . . . . . 7 ⊢ (𝑧 = 𝑏 → (∃𝑦 ∈ 𝑧 𝜑 ↔ ∃𝑦 ∈ 𝑏 𝜑))
20	19	adantl 275	. . . . . 6 ⊢ ((∀𝑥∀𝑦Ⅎ𝑏𝜑 ∧ 𝑧 = 𝑏) → (∃𝑦 ∈ 𝑧 𝜑 ↔ ∃𝑦 ∈ 𝑏 𝜑))
21	18, 20	ralbid 2464	. . . . 5 ⊢ ((∀𝑥∀𝑦Ⅎ𝑏𝜑 ∧ 𝑧 = 𝑏) → (∀𝑥 ∈ 𝑎 ∃𝑦 ∈ 𝑧 𝜑 ↔ ∀𝑥 ∈ 𝑎 ∃𝑦 ∈ 𝑏 𝜑))
22		nfv 1516	. . . . . . 7 ⊢ Ⅎ𝑦 𝑧 = 𝑏
23	8, 22	nfan 1553	. . . . . 6 ⊢ Ⅎ𝑦(∀𝑥∀𝑦Ⅎ𝑏𝜑 ∧ 𝑧 = 𝑏)
24		eleq2 2230	. . . . . . . 8 ⊢ (𝑧 = 𝑏 → (𝑦 ∈ 𝑧 ↔ 𝑦 ∈ 𝑏))
25	24	adantl 275	. . . . . . 7 ⊢ ((∀𝑥∀𝑦Ⅎ𝑏𝜑 ∧ 𝑧 = 𝑏) → (𝑦 ∈ 𝑧 ↔ 𝑦 ∈ 𝑏))
26	25	imbi1d 230	. . . . . 6 ⊢ ((∀𝑥∀𝑦Ⅎ𝑏𝜑 ∧ 𝑧 = 𝑏) → ((𝑦 ∈ 𝑧 → ∃𝑥 ∈ 𝑎 𝜑) ↔ (𝑦 ∈ 𝑏 → ∃𝑥 ∈ 𝑎 𝜑)))
27	23, 26	ralbid2 2470	. . . . 5 ⊢ ((∀𝑥∀𝑦Ⅎ𝑏𝜑 ∧ 𝑧 = 𝑏) → (∀𝑦 ∈ 𝑧 ∃𝑥 ∈ 𝑎 𝜑 ↔ ∀𝑦 ∈ 𝑏 ∃𝑥 ∈ 𝑎 𝜑))
28	21, 27	anbi12d 465	. . . 4 ⊢ ((∀𝑥∀𝑦Ⅎ𝑏𝜑 ∧ 𝑧 = 𝑏) → ((∀𝑥 ∈ 𝑎 ∃𝑦 ∈ 𝑧 𝜑 ∧ ∀𝑦 ∈ 𝑧 ∃𝑥 ∈ 𝑎 𝜑) ↔ (∀𝑥 ∈ 𝑎 ∃𝑦 ∈ 𝑏 𝜑 ∧ ∀𝑦 ∈ 𝑏 ∃𝑥 ∈ 𝑎 𝜑)))
29	28	ex 114	. . 3 ⊢ (∀𝑥∀𝑦Ⅎ𝑏𝜑 → (𝑧 = 𝑏 → ((∀𝑥 ∈ 𝑎 ∃𝑦 ∈ 𝑧 𝜑 ∧ ∀𝑦 ∈ 𝑧 ∃𝑥 ∈ 𝑎 𝜑) ↔ (∀𝑥 ∈ 𝑎 ∃𝑦 ∈ 𝑏 𝜑 ∧ ∀𝑦 ∈ 𝑏 ∃𝑥 ∈ 𝑎 𝜑))))
30	4, 16, 29	cbvexd 1915	. 2 ⊢ (∀𝑥∀𝑦Ⅎ𝑏𝜑 → (∃𝑧(∀𝑥 ∈ 𝑎 ∃𝑦 ∈ 𝑧 𝜑 ∧ ∀𝑦 ∈ 𝑧 ∃𝑥 ∈ 𝑎 𝜑) ↔ ∃𝑏(∀𝑥 ∈ 𝑎 ∃𝑦 ∈ 𝑏 𝜑 ∧ ∀𝑦 ∈ 𝑏 ∃𝑥 ∈ 𝑎 𝜑)))
31	1, 30	syl5ib 153	1 ⊢ (∀𝑥∀𝑦Ⅎ𝑏𝜑 → (∀𝑥 ∈ 𝑎 ∃𝑦𝜑 → ∃𝑏(∀𝑥 ∈ 𝑎 ∃𝑦 ∈ 𝑏 𝜑 ∧ ∀𝑦 ∈ 𝑏 ∃𝑥 ∈ 𝑎 𝜑)))

Syntax hints:   → wi 4   ∧ wa 103   ↔ wb 104  ∀wal 1341  Ⅎwnf 1448  ∃wex 1480  ∀wral 2444  ∃wrex 2445

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-io 699  ax-5 1435  ax-7 1436  ax-gen 1437  ax-ie1 1481  ax-ie2 1482  ax-8 1492  ax-10 1493  ax-11 1494  ax-i12 1495  ax-bndl 1497  ax-4 1498  ax-17 1514  ax-i9 1518  ax-ial 1522  ax-i5r 1523  ax-ext 2147  ax-strcoll 13864

This theorem depends on definitions:  df-bi 116  df-tru 1346  df-nf 1449  df-sb 1751  df-cleq 2158  df-clel 2161  df-nfc 2297  df-ral 2449  df-rex 2450

This theorem is referenced by:  strcollnf  13867