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| Description: Bound-variable hypothesis builder for a class abstraction. Usage of this theorem is discouraged because it depends on ax-13 2376. See nfsab 2726 for a version with more disjoint variable conditions, but not requiring ax-13 2376. (Contributed by Mario Carneiro, 11-Aug-2016.) (New usage is discouraged.) | 
| Ref | Expression | 
|---|---|
| nfsabg.1 | ⊢ Ⅎ𝑥𝜑 | 
| Ref | Expression | 
|---|---|
| nfsabg | ⊢ Ⅎ𝑥 𝑧 ∈ {𝑦 ∣ 𝜑} | 
| Step | Hyp | Ref | Expression | 
|---|---|---|---|
| 1 | nfsabg.1 | . . . 4 ⊢ Ⅎ𝑥𝜑 | |
| 2 | 1 | nf5ri 2194 | . . 3 ⊢ (𝜑 → ∀𝑥𝜑) | 
| 3 | 2 | hbabg 2725 | . 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 ax-13 2376 | 
| This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-tru 1542 df-ex 1779 df-nf 1783 df-sb 2064 df-clab 2714 | 
| This theorem is referenced by: nfabg 2911 | 
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