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| Description: Bound-variable hypothesis builder for a class abstraction. (Contributed by Mario Carneiro, 11-Aug-2016.) Add disjoint variable condition to avoid ax-13 2376. See nfsabg 2727 for a less restrictive version requiring more axioms. (Revised by GG, 20-Jan-2024.) | 
| Ref | Expression | 
|---|---|
| nfsab.1 | ⊢ Ⅎ𝑥𝜑 | 
| Ref | Expression | 
|---|---|
| nfsab | ⊢ Ⅎ𝑥 𝑧 ∈ {𝑦 ∣ 𝜑} | 
| Step | Hyp | Ref | Expression | 
|---|---|---|---|
| 1 | nfsab.1 | . . . 4 ⊢ Ⅎ𝑥𝜑 | |
| 2 | 1 | nf5ri 2194 | . . 3 ⊢ (𝜑 → ∀𝑥𝜑) | 
| 3 | 2 | hbab 2724 | . 2 ⊢ (𝑧 ∈ {𝑦 ∣ 𝜑} → ∀𝑥 𝑧 ∈ {𝑦 ∣ 𝜑}) | 
| 4 | 3 | nf5i 2145 | 1 ⊢ Ⅎ𝑥 𝑧 ∈ {𝑦 ∣ 𝜑} | 
| Colors of variables: wff setvar class | 
| Syntax hints: Ⅎwnf 1782 ∈ wcel 2107 {cab 2713 | 
| This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1794 ax-4 1808 ax-5 1909 ax-6 1966 ax-7 2006 ax-10 2140 ax-11 2156 ax-12 2176 | 
| This theorem depends on definitions: df-bi 207 df-ex 1779 df-nf 1783 df-sb 2064 df-clab 2714 | 
| This theorem is referenced by: nfab 2910 oaun3lem1 43392 upbdrech 45322 ssfiunibd 45326 | 
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