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Mirrors > Home > ILE Home > Th. List > Mathboxes > bdsepnf | GIF version |
Description: Version of ax-bdsep 10675 with one DV condition removed, the other DV condition replaced by a non-freeness hypothesis, and without initial universal quantifier. See also bdsepnfALT 10680. Use bdsep1 10676 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 10678 | . 2 ⊢ (∀𝑥Ⅎ𝑏𝜑 → ∃𝑏∀𝑥(𝑥 ∈ 𝑏 ↔ (𝑥 ∈ 𝑎 ∧ 𝜑))) |
3 | bdsepnf.nf | . 2 ⊢ Ⅎ𝑏𝜑 | |
4 | 2, 3 | mpg 1380 | 1 ⊢ ∃𝑏∀𝑥(𝑥 ∈ 𝑏 ↔ (𝑥 ∈ 𝑎 ∧ 𝜑)) |
Colors of variables: wff set class |
Syntax hints: ∧ wa 102 ↔ wb 103 ∀wal 1282 Ⅎwnf 1389 ∃wex 1421 BOUNDED wbd 10603 |
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 1376 ax-7 1377 ax-gen 1378 ax-ie1 1422 ax-ie2 1423 ax-8 1435 ax-4 1440 ax-14 1445 ax-17 1459 ax-i9 1463 ax-ial 1467 ax-i5r 1468 ax-ext 2063 ax-bdsep 10675 |
This theorem depends on definitions: df-bi 115 df-nf 1390 df-cleq 2074 df-clel 2077 |
This theorem is referenced by: (None) |
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