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| Description: Bound-variable hypothesis builder for a class abstraction. Version of nfabd 2928 with a disjoint variable condition, which does not require ax-13 2377. (Contributed by Mario Carneiro, 8-Oct-2016.) Avoid ax-13 2377. (Revised by GG, 10-Jan-2024.) (Proof shortened by Wolf Lammen, 23-Sep-2024.) | 
| Ref | Expression | 
|---|---|
| nfabdw.1 | ⊢ Ⅎ𝑦𝜑 | 
| nfabdw.2 | ⊢ (𝜑 → Ⅎ𝑥𝜓) | 
| Ref | Expression | 
|---|---|
| nfabdw | ⊢ (𝜑 → Ⅎ𝑥{𝑦 ∣ 𝜓}) | 
| Step | Hyp | Ref | Expression | 
|---|---|---|---|
| 1 | nfv 1914 | . 2 ⊢ Ⅎ𝑧𝜑 | |
| 2 | df-clab 2715 | . . . 4 ⊢ (𝑧 ∈ {𝑦 ∣ 𝜓} ↔ [𝑧 / 𝑦]𝜓) | |
| 3 | sb6 2085 | . . . 4 ⊢ ([𝑧 / 𝑦]𝜓 ↔ ∀𝑦(𝑦 = 𝑧 → 𝜓)) | |
| 4 | 2, 3 | bitri 275 | . . 3 ⊢ (𝑧 ∈ {𝑦 ∣ 𝜓} ↔ ∀𝑦(𝑦 = 𝑧 → 𝜓)) | 
| 5 | nfabdw.1 | . . . 4 ⊢ Ⅎ𝑦𝜑 | |
| 6 | nfvd 1915 | . . . . 5 ⊢ (𝜑 → Ⅎ𝑥 𝑦 = 𝑧) | |
| 7 | nfabdw.2 | . . . . 5 ⊢ (𝜑 → Ⅎ𝑥𝜓) | |
| 8 | 6, 7 | nfimd 1894 | . . . 4 ⊢ (𝜑 → Ⅎ𝑥(𝑦 = 𝑧 → 𝜓)) | 
| 9 | 5, 8 | nfald 2328 | . . 3 ⊢ (𝜑 → Ⅎ𝑥∀𝑦(𝑦 = 𝑧 → 𝜓)) | 
| 10 | 4, 9 | nfxfrd 1854 | . 2 ⊢ (𝜑 → Ⅎ𝑥 𝑧 ∈ {𝑦 ∣ 𝜓}) | 
| 11 | 1, 10 | nfcd 2898 | 1 ⊢ (𝜑 → Ⅎ𝑥{𝑦 ∣ 𝜓}) | 
| Colors of variables: wff setvar class | 
| Syntax hints: → wi 4 ∀wal 1538 Ⅎwnf 1783 [wsb 2064 ∈ wcel 2108 {cab 2714 Ⅎwnfc 2890 | 
| 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-11 2157 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 df-nfc 2892 | 
| This theorem is referenced by: nfrabw 3475 nfrabwOLD 3476 nfsbcdw 3809 nfcsb1d 3921 nfcsbw 3925 nfifd 4555 nfunid 4913 nfopabd 5211 nfiotadw 6517 nfintd 49192 nfiund 49193 | 
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