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Mirrors > Home > ILE Home > Th. List > Mathboxes > bdsep1 | GIF version |
Description: Version of ax-bdsep 10004 without initial universal quantifier. (Contributed by BJ, 5-Oct-2019.) |
Ref | Expression |
---|---|
bdsep1.1 | ⊢ BOUNDED 𝜑 |
Ref | Expression |
---|---|
bdsep1 | ⊢ ∃𝑏∀𝑥(𝑥 ∈ 𝑏 ↔ (𝑥 ∈ 𝑎 ∧ 𝜑)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | bdsep1.1 | . . 3 ⊢ BOUNDED 𝜑 | |
2 | 1 | ax-bdsep 10004 | . 2 ⊢ ∀𝑎∃𝑏∀𝑥(𝑥 ∈ 𝑏 ↔ (𝑥 ∈ 𝑎 ∧ 𝜑)) |
3 | 2 | spi 1429 | 1 ⊢ ∃𝑏∀𝑥(𝑥 ∈ 𝑏 ↔ (𝑥 ∈ 𝑎 ∧ 𝜑)) |
Colors of variables: wff set class |
Syntax hints: ∧ wa 97 ↔ wb 98 ∀wal 1241 ∃wex 1381 BOUNDED wbd 9932 |
This theorem was proved from axioms: ax-mp 7 ax-4 1400 ax-bdsep 10004 |
This theorem is referenced by: bdsep2 10006 bdzfauscl 10010 bdbm1.3ii 10011 bj-axemptylem 10012 bj-nalset 10015 |
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