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Mirrors > Home > ILE Home > Th. List > Mathboxes > bdsep2 | GIF version |
Description: Version of ax-bdsep 9339 with one DV condition removed and without initial universal quantifier. Use bdsep1 9340 when sufficient. (Contributed by BJ, 5-Oct-2019.) |
Ref | Expression |
---|---|
bdsep2.1 | ⊢ BOUNDED φ |
Ref | Expression |
---|---|
bdsep2 | ⊢ ∃𝑏∀x(x ∈ 𝑏 ↔ (x ∈ 𝑎 ∧ φ)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | eleq2 2098 | . . . . . 6 ⊢ (y = 𝑎 → (x ∈ y ↔ x ∈ 𝑎)) | |
2 | 1 | anbi1d 438 | . . . . 5 ⊢ (y = 𝑎 → ((x ∈ y ∧ φ) ↔ (x ∈ 𝑎 ∧ φ))) |
3 | 2 | bibi2d 221 | . . . 4 ⊢ (y = 𝑎 → ((x ∈ 𝑏 ↔ (x ∈ y ∧ φ)) ↔ (x ∈ 𝑏 ↔ (x ∈ 𝑎 ∧ φ)))) |
4 | 3 | albidv 1702 | . . 3 ⊢ (y = 𝑎 → (∀x(x ∈ 𝑏 ↔ (x ∈ y ∧ φ)) ↔ ∀x(x ∈ 𝑏 ↔ (x ∈ 𝑎 ∧ φ)))) |
5 | 4 | exbidv 1703 | . 2 ⊢ (y = 𝑎 → (∃𝑏∀x(x ∈ 𝑏 ↔ (x ∈ y ∧ φ)) ↔ ∃𝑏∀x(x ∈ 𝑏 ↔ (x ∈ 𝑎 ∧ φ)))) |
6 | bdsep2.1 | . . 3 ⊢ BOUNDED φ | |
7 | 6 | bdsep1 9340 | . 2 ⊢ ∃𝑏∀x(x ∈ 𝑏 ↔ (x ∈ y ∧ φ)) |
8 | 5, 7 | chvarv 1809 | 1 ⊢ ∃𝑏∀x(x ∈ 𝑏 ↔ (x ∈ 𝑎 ∧ φ)) |
Colors of variables: wff set class |
Syntax hints: ∧ wa 97 ↔ wb 98 ∀wal 1240 ∃wex 1378 BOUNDED wbd 9267 |
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-5 1333 ax-gen 1335 ax-ie1 1379 ax-ie2 1380 ax-4 1397 ax-17 1416 ax-i9 1420 ax-ial 1424 ax-ext 2019 ax-bdsep 9339 |
This theorem depends on definitions: df-bi 110 df-nf 1347 df-cleq 2030 df-clel 2033 |
This theorem is referenced by: bdsepnft 9342 bdsepnfALT 9344 |
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