Theorem strcollnfALT 16279

Description: Alternate proof of strcollnf 16278, not using strcollnft 16277. (Contributed by BJ, 5-Oct-2019.) (Proof modification is discouraged.) (New usage is discouraged.)

Hypothesis

Ref	Expression
strcollnf.nf	⊢ Ⅎ𝑏𝜑

Assertion

Ref	Expression
strcollnfALT	⊢ (∀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 strcollnfALT

Dummy variable 𝑧 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		strcoll2 16276	. 2 ⊢ (∀𝑥 ∈ 𝑎 ∃𝑦𝜑 → ∃𝑧(∀𝑥 ∈ 𝑎 ∃𝑦 ∈ 𝑧 𝜑 ∧ ∀𝑦 ∈ 𝑧 ∃𝑥 ∈ 𝑎 𝜑))
2		nfcv 2372	. . . . 5 ⊢ Ⅎ𝑏𝑎
3		nfcv 2372	. . . . . 6 ⊢ Ⅎ𝑏𝑧
4		strcollnf.nf	. . . . . 6 ⊢ Ⅎ𝑏𝜑
5	3, 4	nfrexw 2569	. . . . 5 ⊢ Ⅎ𝑏∃𝑦 ∈ 𝑧 𝜑
6	2, 5	nfralxy 2568	. . . 4 ⊢ Ⅎ𝑏∀𝑥 ∈ 𝑎 ∃𝑦 ∈ 𝑧 𝜑
7	2, 4	nfrexw 2569	. . . . 5 ⊢ Ⅎ𝑏∃𝑥 ∈ 𝑎 𝜑
8	3, 7	nfralxy 2568	. . . 4 ⊢ Ⅎ𝑏∀𝑦 ∈ 𝑧 ∃𝑥 ∈ 𝑎 𝜑
9	6, 8	nfan 1611	. . 3 ⊢ Ⅎ𝑏(∀𝑥 ∈ 𝑎 ∃𝑦 ∈ 𝑧 𝜑 ∧ ∀𝑦 ∈ 𝑧 ∃𝑥 ∈ 𝑎 𝜑)
10		nfv 1574	. . . 4 ⊢ Ⅎ𝑧∀𝑥 ∈ 𝑎 ∃𝑦 ∈ 𝑏 𝜑
11		nfv 1574	. . . 4 ⊢ Ⅎ𝑧∀𝑦 ∈ 𝑏 ∃𝑥 ∈ 𝑎 𝜑
12	10, 11	nfan 1611	. . 3 ⊢ Ⅎ𝑧(∀𝑥 ∈ 𝑎 ∃𝑦 ∈ 𝑏 𝜑 ∧ ∀𝑦 ∈ 𝑏 ∃𝑥 ∈ 𝑎 𝜑)
13		rexeq 2729	. . . . 5 ⊢ (𝑧 = 𝑏 → (∃𝑦 ∈ 𝑧 𝜑 ↔ ∃𝑦 ∈ 𝑏 𝜑))
14	13	ralbidv 2530	. . . 4 ⊢ (𝑧 = 𝑏 → (∀𝑥 ∈ 𝑎 ∃𝑦 ∈ 𝑧 𝜑 ↔ ∀𝑥 ∈ 𝑎 ∃𝑦 ∈ 𝑏 𝜑))
15		raleq 2728	. . . 4 ⊢ (𝑧 = 𝑏 → (∀𝑦 ∈ 𝑧 ∃𝑥 ∈ 𝑎 𝜑 ↔ ∀𝑦 ∈ 𝑏 ∃𝑥 ∈ 𝑎 𝜑))
16	14, 15	anbi12d 473	. . 3 ⊢ (𝑧 = 𝑏 → ((∀𝑥 ∈ 𝑎 ∃𝑦 ∈ 𝑧 𝜑 ∧ ∀𝑦 ∈ 𝑧 ∃𝑥 ∈ 𝑎 𝜑) ↔ (∀𝑥 ∈ 𝑎 ∃𝑦 ∈ 𝑏 𝜑 ∧ ∀𝑦 ∈ 𝑏 ∃𝑥 ∈ 𝑎 𝜑)))
17	9, 12, 16	cbvex 1802	. 2 ⊢ (∃𝑧(∀𝑥 ∈ 𝑎 ∃𝑦 ∈ 𝑧 𝜑 ∧ ∀𝑦 ∈ 𝑧 ∃𝑥 ∈ 𝑎 𝜑) ↔ ∃𝑏(∀𝑥 ∈ 𝑎 ∃𝑦 ∈ 𝑏 𝜑 ∧ ∀𝑦 ∈ 𝑏 ∃𝑥 ∈ 𝑎 𝜑))
18	1, 17	sylib 122	1 ⊢ (∀𝑥 ∈ 𝑎 ∃𝑦𝜑 → ∃𝑏(∀𝑥 ∈ 𝑎 ∃𝑦 ∈ 𝑏 𝜑 ∧ ∀𝑦 ∈ 𝑏 ∃𝑥 ∈ 𝑎 𝜑))

Syntax hints:   → wi 4   ∧ wa 104  Ⅎwnf 1506  ∃wex 1538  ∀wral 2508  ∃wrex 2509

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-io 714  ax-5 1493  ax-7 1494  ax-gen 1495  ax-ie1 1539  ax-ie2 1540  ax-8 1550  ax-10 1551  ax-11 1552  ax-i12 1553  ax-bndl 1555  ax-4 1556  ax-17 1572  ax-i9 1576  ax-ial 1580  ax-i5r 1581  ax-ext 2211  ax-strcoll 16275

This theorem depends on definitions:  df-bi 117  df-tru 1398  df-nf 1507  df-sb 1809  df-cleq 2222  df-clel 2225  df-nfc 2361  df-ral 2513  df-rex 2514

This theorem is referenced by: (None)