Theorem strcollnfALT 13173

Description: Alternate proof of strcollnf 13172, not using strcollnft 13171. (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 13170	. 2 ⊢ (∀𝑥 ∈ 𝑎 ∃𝑦𝜑 → ∃𝑧∀𝑦(𝑦 ∈ 𝑧 ↔ ∃𝑥 ∈ 𝑎 𝜑))
2		nfv 1508	. . . . 5 ⊢ Ⅎ𝑏 𝑦 ∈ 𝑧
3		nfcv 2279	. . . . . 6 ⊢ Ⅎ𝑏𝑎
4		strcollnf.nf	. . . . . 6 ⊢ Ⅎ𝑏𝜑
5	3, 4	nfrexxy 2470	. . . . 5 ⊢ Ⅎ𝑏∃𝑥 ∈ 𝑎 𝜑
6	2, 5	nfbi 1568	. . . 4 ⊢ Ⅎ𝑏(𝑦 ∈ 𝑧 ↔ ∃𝑥 ∈ 𝑎 𝜑)
7	6	nfal 1555	. . 3 ⊢ Ⅎ𝑏∀𝑦(𝑦 ∈ 𝑧 ↔ ∃𝑥 ∈ 𝑎 𝜑)
8		nfv 1508	. . 3 ⊢ Ⅎ𝑧∀𝑦(𝑦 ∈ 𝑏 ↔ ∃𝑥 ∈ 𝑎 𝜑)
9		elequ2 1691	. . . . 5 ⊢ (𝑧 = 𝑏 → (𝑦 ∈ 𝑧 ↔ 𝑦 ∈ 𝑏))
10	9	bibi1d 232	. . . 4 ⊢ (𝑧 = 𝑏 → ((𝑦 ∈ 𝑧 ↔ ∃𝑥 ∈ 𝑎 𝜑) ↔ (𝑦 ∈ 𝑏 ↔ ∃𝑥 ∈ 𝑎 𝜑)))
11	10	albidv 1796	. . 3 ⊢ (𝑧 = 𝑏 → (∀𝑦(𝑦 ∈ 𝑧 ↔ ∃𝑥 ∈ 𝑎 𝜑) ↔ ∀𝑦(𝑦 ∈ 𝑏 ↔ ∃𝑥 ∈ 𝑎 𝜑)))
12	7, 8, 11	cbvex 1729	. 2 ⊢ (∃𝑧∀𝑦(𝑦 ∈ 𝑧 ↔ ∃𝑥 ∈ 𝑎 𝜑) ↔ ∃𝑏∀𝑦(𝑦 ∈ 𝑏 ↔ ∃𝑥 ∈ 𝑎 𝜑))
13	1, 12	sylib 121	1 ⊢ (∀𝑥 ∈ 𝑎 ∃𝑦𝜑 → ∃𝑏∀𝑦(𝑦 ∈ 𝑏 ↔ ∃𝑥 ∈ 𝑎 𝜑))

Syntax hints:   → wi 4   ↔ wb 104  ∀wal 1329  Ⅎwnf 1436  ∃wex 1468  ∀wral 2414  ∃wrex 2415

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 698  ax-5 1423  ax-7 1424  ax-gen 1425  ax-ie1 1469  ax-ie2 1470  ax-8 1482  ax-10 1483  ax-11 1484  ax-i12 1485  ax-bndl 1486  ax-4 1487  ax-14 1492  ax-17 1506  ax-i9 1510  ax-ial 1514  ax-i5r 1515  ax-ext 2119  ax-strcoll 13169

This theorem depends on definitions:  df-bi 116  df-tru 1334  df-nf 1437  df-sb 1736  df-cleq 2130  df-clel 2133  df-nfc 2268  df-ral 2419  df-rex 2420

This theorem is referenced by: (None)