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Mirrors > Home > ILE Home > Th. List > Mathboxes > bdsepnf | GIF version |
Description: Version of ax-bdsep 15073 with one disjoint variable condition removed, the other disjoint variable condition replaced by a nonfreeness hypothesis, and without initial universal quantifier. See also bdsepnfALT 15078. Use bdsep1 15074 when sufficient. (Contributed by BJ, 5-Oct-2019.) |
Ref | Expression |
---|---|
bdsepnf.nf | ⊢ Ⅎ𝑏𝜑 |
bdsepnf.1 | ⊢ BOUNDED 𝜑 |
Ref | Expression |
---|---|
bdsepnf | ⊢ ∃𝑏∀𝑥(𝑥 ∈ 𝑏 ↔ (𝑥 ∈ 𝑎 ∧ 𝜑)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | bdsepnf.1 | . . 3 ⊢ BOUNDED 𝜑 | |
2 | 1 | bdsepnft 15076 | . 2 ⊢ (∀𝑥Ⅎ𝑏𝜑 → ∃𝑏∀𝑥(𝑥 ∈ 𝑏 ↔ (𝑥 ∈ 𝑎 ∧ 𝜑))) |
3 | bdsepnf.nf | . 2 ⊢ Ⅎ𝑏𝜑 | |
4 | 2, 3 | mpg 1462 | 1 ⊢ ∃𝑏∀𝑥(𝑥 ∈ 𝑏 ↔ (𝑥 ∈ 𝑎 ∧ 𝜑)) |
Colors of variables: wff set class |
Syntax hints: ∧ wa 104 ↔ wb 105 ∀wal 1362 Ⅎwnf 1471 ∃wex 1503 BOUNDED wbd 15001 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-ia1 106 ax-ia2 107 ax-ia3 108 ax-5 1458 ax-7 1459 ax-gen 1460 ax-ie1 1504 ax-ie2 1505 ax-8 1515 ax-4 1521 ax-17 1537 ax-i9 1541 ax-ial 1545 ax-i5r 1546 ax-14 2163 ax-ext 2171 ax-bdsep 15073 |
This theorem depends on definitions: df-bi 117 df-nf 1472 df-cleq 2182 df-clel 2185 |
This theorem is referenced by: (None) |
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