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| Mirrors > Home > MPE Home > Th. List > hbab1 | Structured version Visualization version GIF version | ||
| Description: Bound-variable hypothesis builder for a class abstraction. (Contributed by NM, 26-May-1993.) (Proof shortened by Wolf Lammen, 25-Oct-2024.) |
| Ref | Expression |
|---|---|
| hbab1 | ⊢ (𝑦 ∈ {𝑥 ∣ 𝜑} → ∀𝑥 𝑦 ∈ {𝑥 ∣ 𝜑}) |
| Step | Hyp | Ref | Expression |
|---|---|---|---|
| 1 | nfsab1 2722 | . 2 ⊢ Ⅎ𝑥 𝑦 ∈ {𝑥 ∣ 𝜑} | |
| 2 | 1 | nf5ri 2195 | 1 ⊢ (𝑦 ∈ {𝑥 ∣ 𝜑} → ∀𝑥 𝑦 ∈ {𝑥 ∣ 𝜑}) |
| Colors of variables: wff setvar class |
| Syntax hints: → wi 4 ∀wal 1538 ∈ 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-6 1967 ax-7 2007 ax-10 2141 ax-12 2177 |
| This theorem depends on definitions: df-bi 207 df-an 396 df-or 849 df-ex 1780 df-nf 1784 df-sb 2065 df-clab 2715 |
| This theorem is referenced by: eqabbOLD 2882 bnj1317 34835 bnj1318 35039 bj-nfsab1 36817 |
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