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Mirrors > Home > ILE Home > Th. List > Mathboxes > strcollnfALT | GIF version |
Description: Alternate proof of strcollnf 9445, not using strcollnft 9444. (Contributed by BJ, 5-Oct-2019.) (Proof modification is discouraged.) (New usage is discouraged.) |
Ref | Expression |
---|---|
strcollnf.nf | ⊢ Ⅎ𝑏φ |
Ref | Expression |
---|---|
strcollnfALT | ⊢ (∀x ∈ 𝑎 ∃yφ → ∃𝑏∀y(y ∈ 𝑏 ↔ ∃x ∈ 𝑎 φ)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | strcoll2 9443 | . 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)) |
Colors of variables: wff set class |
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-bndl 1396 ax-4 1397 ax-14 1402 ax-17 1416 ax-i9 1420 ax-ial 1424 ax-i5r 1425 ax-ext 2019 ax-strcoll 9442 |
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) |
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