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Mirrors > Home > ILE Home > Th. List > Mathboxes > bdsepnf | GIF version |
Description: Version of ax-bdsep 11775 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 11780. Use bdsep1 11776 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 11778 | . 2 ⊢ (∀𝑥Ⅎ𝑏𝜑 → ∃𝑏∀𝑥(𝑥 ∈ 𝑏 ↔ (𝑥 ∈ 𝑎 ∧ 𝜑))) |
3 | bdsepnf.nf | . 2 ⊢ Ⅎ𝑏𝜑 | |
4 | 2, 3 | mpg 1385 | 1 ⊢ ∃𝑏∀𝑥(𝑥 ∈ 𝑏 ↔ (𝑥 ∈ 𝑎 ∧ 𝜑)) |
Colors of variables: wff set class |
Syntax hints: ∧ wa 102 ↔ wb 103 ∀wal 1287 Ⅎwnf 1394 ∃wex 1426 BOUNDED wbd 11703 |
This theorem was proved from axioms: ax-1 5 ax-2 6 ax-mp 7 ax-ia1 104 ax-ia2 105 ax-ia3 106 ax-5 1381 ax-7 1382 ax-gen 1383 ax-ie1 1427 ax-ie2 1428 ax-8 1440 ax-4 1445 ax-14 1450 ax-17 1464 ax-i9 1468 ax-ial 1472 ax-i5r 1473 ax-ext 2070 ax-bdsep 11775 |
This theorem depends on definitions: df-bi 115 df-nf 1395 df-cleq 2081 df-clel 2084 |
This theorem is referenced by: (None) |
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