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| Description: Deduction version of nfsbcw 3810. Version of nfsbcd 3812 with a disjoint variable condition, which does not require ax-13 2377. (Contributed by NM, 23-Nov-2005.) Avoid ax-13 2377. (Revised by GG, 10-Jan-2024.) | 
| Ref | Expression | 
|---|---|
| nfsbcdw.1 | ⊢ Ⅎ𝑦𝜑 | 
| nfsbcdw.2 | ⊢ (𝜑 → Ⅎ𝑥𝐴) | 
| nfsbcdw.3 | ⊢ (𝜑 → Ⅎ𝑥𝜓) | 
| Ref | Expression | 
|---|---|
| nfsbcdw | ⊢ (𝜑 → Ⅎ𝑥[𝐴 / 𝑦]𝜓) | 
| Step | Hyp | Ref | Expression | 
|---|---|---|---|
| 1 | df-sbc 3789 | . 2 ⊢ ([𝐴 / 𝑦]𝜓 ↔ 𝐴 ∈ {𝑦 ∣ 𝜓}) | |
| 2 | nfsbcdw.2 | . . 3 ⊢ (𝜑 → Ⅎ𝑥𝐴) | |
| 3 | nfsbcdw.1 | . . . 4 ⊢ Ⅎ𝑦𝜑 | |
| 4 | nfsbcdw.3 | . . . 4 ⊢ (𝜑 → Ⅎ𝑥𝜓) | |
| 5 | 3, 4 | nfabdw 2927 | . . 3 ⊢ (𝜑 → Ⅎ𝑥{𝑦 ∣ 𝜓}) | 
| 6 | 2, 5 | nfeld 2917 | . 2 ⊢ (𝜑 → Ⅎ𝑥 𝐴 ∈ {𝑦 ∣ 𝜓}) | 
| 7 | 1, 6 | nfxfrd 1854 | 1 ⊢ (𝜑 → Ⅎ𝑥[𝐴 / 𝑦]𝜓) | 
| Colors of variables: wff setvar class | 
| Syntax hints: → wi 4 Ⅎwnf 1783 ∈ wcel 2108 {cab 2714 Ⅎwnfc 2890 [wsbc 3788 | 
| 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-8 2110 ax-9 2118 ax-10 2141 ax-11 2157 ax-12 2177 ax-ext 2708 | 
| 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-cleq 2729 df-clel 2816 df-nfc 2892 df-sbc 3789 | 
| This theorem is referenced by: nfsbcw 3810 nfcsbw 3925 sbcnestgfw 4421 | 
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