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Mirrors > Home > ILE Home > Th. List > Mathboxes > bdsepnf | GIF version |
Description: Version of ax-bdsep 13082 with one disjoint variable condition removed, the other disjoint variable condition replaced by a non-freeness hypothesis, and without initial universal quantifier. See also bdsepnfALT 13087. Use bdsep1 13083 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 13085 | . 2 ⊢ (∀𝑥Ⅎ𝑏𝜑 → ∃𝑏∀𝑥(𝑥 ∈ 𝑏 ↔ (𝑥 ∈ 𝑎 ∧ 𝜑))) |
3 | bdsepnf.nf | . 2 ⊢ Ⅎ𝑏𝜑 | |
4 | 2, 3 | mpg 1427 | 1 ⊢ ∃𝑏∀𝑥(𝑥 ∈ 𝑏 ↔ (𝑥 ∈ 𝑎 ∧ 𝜑)) |
Colors of variables: wff set class |
Syntax hints: ∧ wa 103 ↔ wb 104 ∀wal 1329 Ⅎwnf 1436 ∃wex 1468 BOUNDED wbd 13010 |
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 1423 ax-7 1424 ax-gen 1425 ax-ie1 1469 ax-ie2 1470 ax-8 1482 ax-4 1487 ax-14 1492 ax-17 1506 ax-i9 1510 ax-ial 1514 ax-i5r 1515 ax-ext 2121 ax-bdsep 13082 |
This theorem depends on definitions: df-bi 116 df-nf 1437 df-cleq 2132 df-clel 2135 |
This theorem is referenced by: (None) |
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