Theorem strcollnfALT 10781

Description: Alternate proof of strcollnf 10780, not using strcollnft 10779. (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∀𝑦(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 10778	. 2 ⊢ (∀𝑥 ∈ 𝑎 ∃𝑦𝜑 → ∃𝑧∀𝑦(𝑦 ∈ 𝑧 ↔ ∃𝑥 ∈ 𝑎 𝜑))
2		nfv 1461	. . . . 5 ⊢ Ⅎ𝑏 𝑦 ∈ 𝑧
3		nfcv 2219	. . . . . 6 ⊢ Ⅎ𝑏𝑎
4		strcollnf.nf	. . . . . 6 ⊢ Ⅎ𝑏𝜑
5	3, 4	nfrexxy 2403	. . . . 5 ⊢ Ⅎ𝑏∃𝑥 ∈ 𝑎 𝜑
6	2, 5	nfbi 1521	. . . 4 ⊢ Ⅎ𝑏(𝑦 ∈ 𝑧 ↔ ∃𝑥 ∈ 𝑎 𝜑)
7	6	nfal 1508	. . 3 ⊢ Ⅎ𝑏∀𝑦(𝑦 ∈ 𝑧 ↔ ∃𝑥 ∈ 𝑎 𝜑)
8		nfv 1461	. . 3 ⊢ Ⅎ𝑧∀𝑦(𝑦 ∈ 𝑏 ↔ ∃𝑥 ∈ 𝑎 𝜑)
9		elequ2 1641	. . . . 5 ⊢ (𝑧 = 𝑏 → (𝑦 ∈ 𝑧 ↔ 𝑦 ∈ 𝑏))
10	9	bibi1d 231	. . . 4 ⊢ (𝑧 = 𝑏 → ((𝑦 ∈ 𝑧 ↔ ∃𝑥 ∈ 𝑎 𝜑) ↔ (𝑦 ∈ 𝑏 ↔ ∃𝑥 ∈ 𝑎 𝜑)))
11	10	albidv 1745	. . 3 ⊢ (𝑧 = 𝑏 → (∀𝑦(𝑦 ∈ 𝑧 ↔ ∃𝑥 ∈ 𝑎 𝜑) ↔ ∀𝑦(𝑦 ∈ 𝑏 ↔ ∃𝑥 ∈ 𝑎 𝜑)))
12	7, 8, 11	cbvex 1679	. 2 ⊢ (∃𝑧∀𝑦(𝑦 ∈ 𝑧 ↔ ∃𝑥 ∈ 𝑎 𝜑) ↔ ∃𝑏∀𝑦(𝑦 ∈ 𝑏 ↔ ∃𝑥 ∈ 𝑎 𝜑))
13	1, 12	sylib 120	1 ⊢ (∀𝑥 ∈ 𝑎 ∃𝑦𝜑 → ∃𝑏∀𝑦(𝑦 ∈ 𝑏 ↔ ∃𝑥 ∈ 𝑎 𝜑))

Syntax hints:   → wi 4   ↔ wb 103  ∀wal 1282  Ⅎwnf 1389  ∃wex 1421  ∀wral 2348  ∃wrex 2349

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106  ax-io 662  ax-5 1376  ax-7 1377  ax-gen 1378  ax-ie1 1422  ax-ie2 1423  ax-8 1435  ax-10 1436  ax-11 1437  ax-i12 1438  ax-bndl 1439  ax-4 1440  ax-14 1445  ax-17 1459  ax-i9 1463  ax-ial 1467  ax-i5r 1468  ax-ext 2063  ax-strcoll 10777

This theorem depends on definitions:  df-bi 115  df-tru 1287  df-nf 1390  df-sb 1686  df-cleq 2074  df-clel 2077  df-nfc 2208  df-ral 2353  df-rex 2354

This theorem is referenced by: (None)