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| Mirrors > Home > MPE Home > Th. List > hbab | Structured version Visualization version GIF version | ||
| Description: Bound-variable hypothesis builder for a class abstraction. (Contributed by NM, 1-Mar-1995.) Add disjoint variable condition to avoid ax-13 2377. See hbabg 2726 for a less restrictive version requiring more axioms. (Revised by GG, 20-Jan-2024.) | 
| Ref | Expression | 
|---|---|
| hbab.1 | ⊢ (𝜑 → ∀𝑥𝜑) | 
| Ref | Expression | 
|---|---|
| hbab | ⊢ (𝑧 ∈ {𝑦 ∣ 𝜑} → ∀𝑥 𝑧 ∈ {𝑦 ∣ 𝜑}) | 
| Step | Hyp | Ref | Expression | 
|---|---|---|---|
| 1 | df-clab 2715 | . 2 ⊢ (𝑧 ∈ {𝑦 ∣ 𝜑} ↔ [𝑧 / 𝑦]𝜑) | |
| 2 | hbab.1 | . . 3 ⊢ (𝜑 → ∀𝑥𝜑) | |
| 3 | 2 | hbsbw 2171 | . 2 ⊢ ([𝑧 / 𝑦]𝜑 → ∀𝑥[𝑧 / 𝑦]𝜑) | 
| 4 | 1, 3 | hbxfrbi 1825 | 1 ⊢ (𝑧 ∈ {𝑦 ∣ 𝜑} → ∀𝑥 𝑧 ∈ {𝑦 ∣ 𝜑}) | 
| Colors of variables: wff setvar class | 
| Syntax hints: → wi 4 ∀wal 1538 [wsb 2064 ∈ wcel 2108 {cab 2714 | 
| This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1795 ax-4 1809 ax-5 1910 ax-11 2157 | 
| This theorem depends on definitions: df-bi 207 df-ex 1780 df-sb 2065 df-clab 2715 | 
| This theorem is referenced by: nfsab 2727 bnj1441 34854 bnj1309 35036 | 
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