Theorem strcollnfALT 16056

Description: Alternate proof of strcollnf 16055, not using strcollnft 16054. (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 16053	. 2 ⊢ (∀𝑥 ∈ 𝑎 ∃𝑦𝜑 → ∃𝑧(∀𝑥 ∈ 𝑎 ∃𝑦 ∈ 𝑧 𝜑 ∧ ∀𝑦 ∈ 𝑧 ∃𝑥 ∈ 𝑎 𝜑))
2		nfcv 2349	. . . . 5 ⊢ Ⅎ𝑏𝑎
3		nfcv 2349	. . . . . 6 ⊢ Ⅎ𝑏𝑧
4		strcollnf.nf	. . . . . 6 ⊢ Ⅎ𝑏𝜑
5	3, 4	nfrexw 2546	. . . . 5 ⊢ Ⅎ𝑏∃𝑦 ∈ 𝑧 𝜑
6	2, 5	nfralxy 2545	. . . 4 ⊢ Ⅎ𝑏∀𝑥 ∈ 𝑎 ∃𝑦 ∈ 𝑧 𝜑
7	2, 4	nfrexw 2546	. . . . 5 ⊢ Ⅎ𝑏∃𝑥 ∈ 𝑎 𝜑
8	3, 7	nfralxy 2545	. . . 4 ⊢ Ⅎ𝑏∀𝑦 ∈ 𝑧 ∃𝑥 ∈ 𝑎 𝜑
9	6, 8	nfan 1589	. . 3 ⊢ Ⅎ𝑏(∀𝑥 ∈ 𝑎 ∃𝑦 ∈ 𝑧 𝜑 ∧ ∀𝑦 ∈ 𝑧 ∃𝑥 ∈ 𝑎 𝜑)
10		nfv 1552	. . . 4 ⊢ Ⅎ𝑧∀𝑥 ∈ 𝑎 ∃𝑦 ∈ 𝑏 𝜑
11		nfv 1552	. . . 4 ⊢ Ⅎ𝑧∀𝑦 ∈ 𝑏 ∃𝑥 ∈ 𝑎 𝜑
12	10, 11	nfan 1589	. . 3 ⊢ Ⅎ𝑧(∀𝑥 ∈ 𝑎 ∃𝑦 ∈ 𝑏 𝜑 ∧ ∀𝑦 ∈ 𝑏 ∃𝑥 ∈ 𝑎 𝜑)
13		rexeq 2704	. . . . 5 ⊢ (𝑧 = 𝑏 → (∃𝑦 ∈ 𝑧 𝜑 ↔ ∃𝑦 ∈ 𝑏 𝜑))
14	13	ralbidv 2507	. . . 4 ⊢ (𝑧 = 𝑏 → (∀𝑥 ∈ 𝑎 ∃𝑦 ∈ 𝑧 𝜑 ↔ ∀𝑥 ∈ 𝑎 ∃𝑦 ∈ 𝑏 𝜑))
15		raleq 2703	. . . 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 1780	. 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 1484  ∃wex 1516  ∀wral 2485  ∃wrex 2486

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 711  ax-5 1471  ax-7 1472  ax-gen 1473  ax-ie1 1517  ax-ie2 1518  ax-8 1528  ax-10 1529  ax-11 1530  ax-i12 1531  ax-bndl 1533  ax-4 1534  ax-17 1550  ax-i9 1554  ax-ial 1558  ax-i5r 1559  ax-ext 2188  ax-strcoll 16052

This theorem depends on definitions:  df-bi 117  df-tru 1376  df-nf 1485  df-sb 1787  df-cleq 2199  df-clel 2202  df-nfc 2338  df-ral 2490  df-rex 2491

This theorem is referenced by: (None)