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Mirrors > Home > ILE Home > Th. List > nfsbc1d | GIF version |
Description: Deduction version of nfsbc1 2899. (Contributed by NM, 23-May-2006.) (Revised by Mario Carneiro, 12-Oct-2016.) |
Ref | Expression |
---|---|
nfsbc1d.2 | ⊢ (𝜑 → Ⅎ𝑥𝐴) |
Ref | Expression |
---|---|
nfsbc1d | ⊢ (𝜑 → Ⅎ𝑥[𝐴 / 𝑥]𝜓) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | df-sbc 2883 | . 2 ⊢ ([𝐴 / 𝑥]𝜓 ↔ 𝐴 ∈ {𝑥 ∣ 𝜓}) | |
2 | nfsbc1d.2 | . . 3 ⊢ (𝜑 → Ⅎ𝑥𝐴) | |
3 | nfab1 2260 | . . . 4 ⊢ Ⅎ𝑥{𝑥 ∣ 𝜓} | |
4 | 3 | a1i 9 | . . 3 ⊢ (𝜑 → Ⅎ𝑥{𝑥 ∣ 𝜓}) |
5 | 2, 4 | nfeld 2274 | . 2 ⊢ (𝜑 → Ⅎ𝑥 𝐴 ∈ {𝑥 ∣ 𝜓}) |
6 | 1, 5 | nfxfrd 1436 | 1 ⊢ (𝜑 → Ⅎ𝑥[𝐴 / 𝑥]𝜓) |
Colors of variables: wff set class |
Syntax hints: → wi 4 Ⅎwnf 1421 ∈ wcel 1465 {cab 2103 Ⅎwnfc 2245 [wsbc 2882 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-ia1 105 ax-ia2 106 ax-ia3 107 ax-5 1408 ax-7 1409 ax-gen 1410 ax-ie1 1454 ax-ie2 1455 ax-8 1467 ax-11 1469 ax-4 1472 ax-17 1491 ax-i9 1495 ax-ial 1499 ax-i5r 1500 ax-ext 2099 |
This theorem depends on definitions: df-bi 116 df-nf 1422 df-sb 1721 df-clab 2104 df-cleq 2110 df-clel 2113 df-nfc 2247 df-sbc 2883 |
This theorem is referenced by: nfsbc1 2899 nfcsb1d 3003 |
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