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Mirrors > Home > ILE Home > Th. List > Mathboxes > bdsepnfALT | GIF version |
Description: Alternate proof of bdsepnf 13013, not using bdsepnft 13012. (Contributed by BJ, 5-Oct-2019.) (Proof modification is discouraged.) (New usage is discouraged.) |
Ref | Expression |
---|---|
bdsepnf.nf | ⊢ Ⅎ𝑏𝜑 |
bdsepnf.1 | ⊢ BOUNDED 𝜑 |
Ref | Expression |
---|---|
bdsepnfALT | ⊢ ∃𝑏∀𝑥(𝑥 ∈ 𝑏 ↔ (𝑥 ∈ 𝑎 ∧ 𝜑)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | bdsepnf.1 | . . 3 ⊢ BOUNDED 𝜑 | |
2 | 1 | bdsep2 13011 | . 2 ⊢ ∃𝑦∀𝑥(𝑥 ∈ 𝑦 ↔ (𝑥 ∈ 𝑎 ∧ 𝜑)) |
3 | nfv 1493 | . . . . 5 ⊢ Ⅎ𝑏 𝑥 ∈ 𝑦 | |
4 | nfv 1493 | . . . . . 6 ⊢ Ⅎ𝑏 𝑥 ∈ 𝑎 | |
5 | bdsepnf.nf | . . . . . 6 ⊢ Ⅎ𝑏𝜑 | |
6 | 4, 5 | nfan 1529 | . . . . 5 ⊢ Ⅎ𝑏(𝑥 ∈ 𝑎 ∧ 𝜑) |
7 | 3, 6 | nfbi 1553 | . . . 4 ⊢ Ⅎ𝑏(𝑥 ∈ 𝑦 ↔ (𝑥 ∈ 𝑎 ∧ 𝜑)) |
8 | 7 | nfal 1540 | . . 3 ⊢ Ⅎ𝑏∀𝑥(𝑥 ∈ 𝑦 ↔ (𝑥 ∈ 𝑎 ∧ 𝜑)) |
9 | nfv 1493 | . . 3 ⊢ Ⅎ𝑦∀𝑥(𝑥 ∈ 𝑏 ↔ (𝑥 ∈ 𝑎 ∧ 𝜑)) | |
10 | elequ2 1676 | . . . . 5 ⊢ (𝑦 = 𝑏 → (𝑥 ∈ 𝑦 ↔ 𝑥 ∈ 𝑏)) | |
11 | 10 | bibi1d 232 | . . . 4 ⊢ (𝑦 = 𝑏 → ((𝑥 ∈ 𝑦 ↔ (𝑥 ∈ 𝑎 ∧ 𝜑)) ↔ (𝑥 ∈ 𝑏 ↔ (𝑥 ∈ 𝑎 ∧ 𝜑)))) |
12 | 11 | albidv 1780 | . . 3 ⊢ (𝑦 = 𝑏 → (∀𝑥(𝑥 ∈ 𝑦 ↔ (𝑥 ∈ 𝑎 ∧ 𝜑)) ↔ ∀𝑥(𝑥 ∈ 𝑏 ↔ (𝑥 ∈ 𝑎 ∧ 𝜑)))) |
13 | 8, 9, 12 | cbvex 1714 | . 2 ⊢ (∃𝑦∀𝑥(𝑥 ∈ 𝑦 ↔ (𝑥 ∈ 𝑎 ∧ 𝜑)) ↔ ∃𝑏∀𝑥(𝑥 ∈ 𝑏 ↔ (𝑥 ∈ 𝑎 ∧ 𝜑))) |
14 | 2, 13 | mpbi 144 | 1 ⊢ ∃𝑏∀𝑥(𝑥 ∈ 𝑏 ↔ (𝑥 ∈ 𝑎 ∧ 𝜑)) |
Colors of variables: wff set class |
Syntax hints: ∧ wa 103 ↔ wb 104 ∀wal 1314 Ⅎwnf 1421 ∃wex 1453 BOUNDED wbd 12937 |
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-5 1408 ax-7 1409 ax-gen 1410 ax-ie1 1454 ax-ie2 1455 ax-8 1467 ax-4 1472 ax-14 1477 ax-17 1491 ax-i9 1495 ax-ial 1499 ax-i5r 1500 ax-ext 2099 ax-bdsep 13009 |
This theorem depends on definitions: df-bi 116 df-tru 1319 df-nf 1422 df-cleq 2110 df-clel 2113 |
This theorem is referenced by: (None) |
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