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Mirrors > Home > ILE Home > Th. List > Mathboxes > bdph | GIF version |
Description: A formula which defines (by class abstraction) a bounded class is bounded. (Contributed by BJ, 6-Oct-2019.) |
Ref | Expression |
---|---|
bdph.1 | ⊢ BOUNDED {𝑥 ∣ 𝜑} |
Ref | Expression |
---|---|
bdph | ⊢ BOUNDED 𝜑 |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | bdph.1 | . . . . 5 ⊢ BOUNDED {𝑥 ∣ 𝜑} | |
2 | 1 | bdeli 15036 | . . . 4 ⊢ BOUNDED 𝑦 ∈ {𝑥 ∣ 𝜑} |
3 | df-clab 2176 | . . . 4 ⊢ (𝑦 ∈ {𝑥 ∣ 𝜑} ↔ [𝑦 / 𝑥]𝜑) | |
4 | 2, 3 | bd0 15014 | . . 3 ⊢ BOUNDED [𝑦 / 𝑥]𝜑 |
5 | 4 | ax-bdsb 15012 | . 2 ⊢ BOUNDED [𝑥 / 𝑦][𝑦 / 𝑥]𝜑 |
6 | sbid2v 2008 | . 2 ⊢ ([𝑥 / 𝑦][𝑦 / 𝑥]𝜑 ↔ 𝜑) | |
7 | 5, 6 | bd0 15014 | 1 ⊢ BOUNDED 𝜑 |
Colors of variables: wff set class |
Syntax hints: [wsb 1773 ∈ wcel 2160 {cab 2175 BOUNDED wbd 15002 BOUNDED wbdc 15030 |
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-gen 1460 ax-ie1 1504 ax-ie2 1505 ax-8 1515 ax-11 1517 ax-4 1521 ax-17 1537 ax-i9 1541 ax-ial 1545 ax-bd0 15003 ax-bdsb 15012 |
This theorem depends on definitions: df-bi 117 df-sb 1774 df-clab 2176 df-bdc 15031 |
This theorem is referenced by: bds 15041 |
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