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Mirrors > Home > ILE Home > Th. List > Mathboxes > bdsep2 | GIF version |
Description: Version of ax-bdsep 13601 with one disjoint variable condition removed and without initial universal quantifier. Use bdsep1 13602 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 2228 | . . . . . 6 ⊢ (𝑦 = 𝑎 → (𝑥 ∈ 𝑦 ↔ 𝑥 ∈ 𝑎)) | |
2 | 1 | anbi1d 461 | . . . . 5 ⊢ (𝑦 = 𝑎 → ((𝑥 ∈ 𝑦 ∧ 𝜑) ↔ (𝑥 ∈ 𝑎 ∧ 𝜑))) |
3 | 2 | bibi2d 231 | . . . 4 ⊢ (𝑦 = 𝑎 → ((𝑥 ∈ 𝑏 ↔ (𝑥 ∈ 𝑦 ∧ 𝜑)) ↔ (𝑥 ∈ 𝑏 ↔ (𝑥 ∈ 𝑎 ∧ 𝜑)))) |
4 | 3 | albidv 1811 | . . 3 ⊢ (𝑦 = 𝑎 → (∀𝑥(𝑥 ∈ 𝑏 ↔ (𝑥 ∈ 𝑦 ∧ 𝜑)) ↔ ∀𝑥(𝑥 ∈ 𝑏 ↔ (𝑥 ∈ 𝑎 ∧ 𝜑)))) |
5 | 4 | exbidv 1812 | . 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 13602 | . 2 ⊢ ∃𝑏∀𝑥(𝑥 ∈ 𝑏 ↔ (𝑥 ∈ 𝑦 ∧ 𝜑)) |
8 | 5, 7 | chvarv 1924 | 1 ⊢ ∃𝑏∀𝑥(𝑥 ∈ 𝑏 ↔ (𝑥 ∈ 𝑎 ∧ 𝜑)) |
Colors of variables: wff set class |
Syntax hints: ∧ wa 103 ↔ wb 104 ∀wal 1340 ∃wex 1479 BOUNDED wbd 13529 |
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 1434 ax-gen 1436 ax-ie1 1480 ax-ie2 1481 ax-4 1497 ax-17 1513 ax-i9 1517 ax-ial 1521 ax-ext 2146 ax-bdsep 13601 |
This theorem depends on definitions: df-bi 116 df-nf 1448 df-cleq 2157 df-clel 2160 |
This theorem is referenced by: bdsepnft 13604 bdsepnfALT 13606 |
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