|   | Metamath Proof Explorer | < Previous  
      Next > Nearby theorems | |
| Mirrors > Home > MPE Home > Th. List > nfsbcdw | Structured version Visualization version GIF version | ||
| Description: Deduction version of nfsbcw 3809. Version of nfsbcd 3811 with a disjoint variable condition, which does not require ax-13 2376. (Contributed by NM, 23-Nov-2005.) Avoid ax-13 2376. (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 3788 | . 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 2926 | . . 3 ⊢ (𝜑 → Ⅎ𝑥{𝑦 ∣ 𝜓}) | 
| 6 | 2, 5 | nfeld 2916 | . 2 ⊢ (𝜑 → Ⅎ𝑥 𝐴 ∈ {𝑦 ∣ 𝜓}) | 
| 7 | 1, 6 | nfxfrd 1853 | 1 ⊢ (𝜑 → Ⅎ𝑥[𝐴 / 𝑦]𝜓) | 
| Colors of variables: wff setvar class | 
| Syntax hints: → wi 4 Ⅎwnf 1782 ∈ wcel 2107 {cab 2713 Ⅎwnfc 2889 [wsbc 3787 | 
| 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-8 2109 ax-9 2117 ax-10 2140 ax-11 2156 ax-12 2176 ax-ext 2707 | 
| This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-ex 1779 df-nf 1783 df-sb 2064 df-clab 2714 df-cleq 2728 df-clel 2815 df-nfc 2891 df-sbc 3788 | 
| This theorem is referenced by: nfsbcw 3809 nfcsbw 3924 sbcnestgfw 4420 | 
| Copyright terms: Public domain | W3C validator |