Theorem strcollnfALT 9416

Description: Alternate proof of strcollnf 9415, not using strcollnft 9414. (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 9413	. 2 ⊢ (∀x ∈ 𝑎 ∃yφ → ∃z∀y(y ∈ z ↔ ∃x ∈ 𝑎 φ))
2		nfv 1418	. . . . 5 ⊢ Ⅎ𝑏 y ∈ z
3		nfcv 2175	. . . . . 6 ⊢ Ⅎ𝑏𝑎
4		strcollnf.nf	. . . . . 6 ⊢ Ⅎ𝑏φ
5	3, 4	nfrexxy 2355	. . . . 5 ⊢ Ⅎ𝑏∃x ∈ 𝑎 φ
6	2, 5	nfbi 1478	. . . 4 ⊢ Ⅎ𝑏(y ∈ z ↔ ∃x ∈ 𝑎 φ)
7	6	nfal 1465	. . 3 ⊢ Ⅎ𝑏∀y(y ∈ z ↔ ∃x ∈ 𝑎 φ)
8		nfv 1418	. . 3 ⊢ Ⅎz∀y(y ∈ 𝑏 ↔ ∃x ∈ 𝑎 φ)
9		elequ2 1598	. . . . 5 ⊢ (z = 𝑏 → (y ∈ z ↔ y ∈ 𝑏))
10	9	bibi1d 222	. . . 4 ⊢ (z = 𝑏 → ((y ∈ z ↔ ∃x ∈ 𝑎 φ) ↔ (y ∈ 𝑏 ↔ ∃x ∈ 𝑎 φ)))
11	10	albidv 1702	. . 3 ⊢ (z = 𝑏 → (∀y(y ∈ z ↔ ∃x ∈ 𝑎 φ) ↔ ∀y(y ∈ 𝑏 ↔ ∃x ∈ 𝑎 φ)))
12	7, 8, 11	cbvex 1636	. 2 ⊢ (∃z∀y(y ∈ z ↔ ∃x ∈ 𝑎 φ) ↔ ∃𝑏∀y(y ∈ 𝑏 ↔ ∃x ∈ 𝑎 φ))
13	1, 12	sylib 127	1 ⊢ (∀x ∈ 𝑎 ∃yφ → ∃𝑏∀y(y ∈ 𝑏 ↔ ∃x ∈ 𝑎 φ))

Syntax hints:   → wi 4   ↔ wb 98  ∀wal 1240  Ⅎwnf 1346  ∃wex 1378  ∀wral 2300  ∃wrex 2301

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 99  ax-ia2 100  ax-ia3 101  ax-io 629  ax-5 1333  ax-7 1334  ax-gen 1335  ax-ie1 1379  ax-ie2 1380  ax-8 1392  ax-10 1393  ax-11 1394  ax-i12 1395  ax-bnd 1396  ax-4 1397  ax-14 1402  ax-17 1416  ax-i9 1420  ax-ial 1424  ax-i5r 1425  ax-ext 2019  ax-strcoll 9412

This theorem depends on definitions:  df-bi 110  df-tru 1245  df-nf 1347  df-sb 1643  df-cleq 2030  df-clel 2033  df-nfc 2164  df-ral 2305  df-rex 2306

This theorem is referenced by: (None)