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Mirrors > Home > ILE Home > Th. List > Mathboxes > bdsep2 | GIF version |
Description: Version of ax-bdsep 10004 with one DV condition removed and without initial universal quantifier. Use bdsep1 10005 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 2101 | . . . . . 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 1705 | . . 3 ⊢ (𝑦 = 𝑎 → (∀𝑥(𝑥 ∈ 𝑏 ↔ (𝑥 ∈ 𝑦 ∧ 𝜑)) ↔ ∀𝑥(𝑥 ∈ 𝑏 ↔ (𝑥 ∈ 𝑎 ∧ 𝜑)))) |
5 | 4 | exbidv 1706 | . 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 10005 | . 2 ⊢ ∃𝑏∀𝑥(𝑥 ∈ 𝑏 ↔ (𝑥 ∈ 𝑦 ∧ 𝜑)) |
8 | 5, 7 | chvarv 1812 | 1 ⊢ ∃𝑏∀𝑥(𝑥 ∈ 𝑏 ↔ (𝑥 ∈ 𝑎 ∧ 𝜑)) |
Colors of variables: wff set class |
Syntax hints: ∧ wa 97 ↔ wb 98 ∀wal 1241 ∃wex 1381 BOUNDED wbd 9932 |
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 1336 ax-gen 1338 ax-ie1 1382 ax-ie2 1383 ax-4 1400 ax-17 1419 ax-i9 1423 ax-ial 1427 ax-ext 2022 ax-bdsep 10004 |
This theorem depends on definitions: df-bi 110 df-nf 1350 df-cleq 2033 df-clel 2036 |
This theorem is referenced by: bdsepnft 10007 bdsepnfALT 10009 |
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