Theorem strcollnfALT 16995

Description: Alternate proof of strcollnf 16994, not using strcollnft 16993. (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 16992	. 2 ⊢ (∀𝑥 ∈ 𝑎 ∃𝑦𝜑 → ∃𝑧(∀𝑥 ∈ 𝑎 ∃𝑦 ∈ 𝑧 𝜑 ∧ ∀𝑦 ∈ 𝑧 ∃𝑥 ∈ 𝑎 𝜑))
2		nfcv 2392	. . . . 5 ⊢ Ⅎ𝑏𝑎
3		nfcv 2392	. . . . . 6 ⊢ Ⅎ𝑏𝑧
4		strcollnf.nf	. . . . . 6 ⊢ Ⅎ𝑏𝜑
5	3, 4	nfrexw 2589	. . . . 5 ⊢ Ⅎ𝑏∃𝑦 ∈ 𝑧 𝜑
6	2, 5	nfralxy 2588	. . . 4 ⊢ Ⅎ𝑏∀𝑥 ∈ 𝑎 ∃𝑦 ∈ 𝑧 𝜑
7	2, 4	nfrexw 2589	. . . . 5 ⊢ Ⅎ𝑏∃𝑥 ∈ 𝑎 𝜑
8	3, 7	nfralxy 2588	. . . 4 ⊢ Ⅎ𝑏∀𝑦 ∈ 𝑧 ∃𝑥 ∈ 𝑎 𝜑
9	6, 8	nfan 1618	. . 3 ⊢ Ⅎ𝑏(∀𝑥 ∈ 𝑎 ∃𝑦 ∈ 𝑧 𝜑 ∧ ∀𝑦 ∈ 𝑧 ∃𝑥 ∈ 𝑎 𝜑)
10		nfv 1581	. . . 4 ⊢ Ⅎ𝑧∀𝑥 ∈ 𝑎 ∃𝑦 ∈ 𝑏 𝜑
11		nfv 1581	. . . 4 ⊢ Ⅎ𝑧∀𝑦 ∈ 𝑏 ∃𝑥 ∈ 𝑎 𝜑
12	10, 11	nfan 1618	. . 3 ⊢ Ⅎ𝑧(∀𝑥 ∈ 𝑎 ∃𝑦 ∈ 𝑏 𝜑 ∧ ∀𝑦 ∈ 𝑏 ∃𝑥 ∈ 𝑎 𝜑)
13		rexeq 2750	. . . . 5 ⊢ (𝑧 = 𝑏 → (∃𝑦 ∈ 𝑧 𝜑 ↔ ∃𝑦 ∈ 𝑏 𝜑))
14	13	ralbidv 2550	. . . 4 ⊢ (𝑧 = 𝑏 → (∀𝑥 ∈ 𝑎 ∃𝑦 ∈ 𝑧 𝜑 ↔ ∀𝑥 ∈ 𝑎 ∃𝑦 ∈ 𝑏 𝜑))
15		raleq 2749	. . . 4 ⊢ (𝑧 = 𝑏 → (∀𝑦 ∈ 𝑧 ∃𝑥 ∈ 𝑎 𝜑 ↔ ∀𝑦 ∈ 𝑏 ∃𝑥 ∈ 𝑎 𝜑))
16	14, 15	anbi12d 477	. . 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 1809	. 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 1513  ∃wex 1545  ∀wral 2528  ∃wrex 2529

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 721  ax-5 1500  ax-7 1501  ax-gen 1502  ax-ie1 1546  ax-ie2 1547  ax-8 1557  ax-10 1558  ax-11 1559  ax-i12 1560  ax-bndl 1562  ax-4 1563  ax-17 1579  ax-i9 1583  ax-ial 1587  ax-i5r 1588  ax-ext 2220  ax-strcoll 16991

This theorem depends on definitions:  df-bi 117  df-tru 1405  df-nf 1514  df-sb 1816  df-cleq 2231  df-clel 2234  df-nfc 2381  df-ral 2533  df-rex 2534

This theorem is referenced by: (None)