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Mirrors > Home > ILE Home > Th. List > Mathboxes > bdab | GIF version |
Description: Membership in a class defined by class abstraction using a bounded formula, is a bounded formula. (Contributed by BJ, 3-Oct-2019.) |
Ref | Expression |
---|---|
bdab.1 | ⊢ BOUNDED 𝜑 |
Ref | Expression |
---|---|
bdab | ⊢ BOUNDED 𝑥 ∈ {𝑦 ∣ 𝜑} |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | bdab.1 | . . 3 ⊢ BOUNDED 𝜑 | |
2 | 1 | ax-bdsb 15027 | . 2 ⊢ BOUNDED [𝑥 / 𝑦]𝜑 |
3 | df-clab 2176 | . 2 ⊢ (𝑥 ∈ {𝑦 ∣ 𝜑} ↔ [𝑥 / 𝑦]𝜑) | |
4 | 2, 3 | bd0r 15030 | 1 ⊢ BOUNDED 𝑥 ∈ {𝑦 ∣ 𝜑} |
Colors of variables: wff set class |
Syntax hints: [wsb 1773 ∈ wcel 2160 {cab 2175 BOUNDED wbd 15017 |
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-bd0 15018 ax-bdsb 15027 |
This theorem depends on definitions: df-bi 117 df-clab 2176 |
This theorem is referenced by: bdcab 15054 bdsbcALT 15064 |
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