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Mirrors > Home > ILE Home > Th. List > nfabd | GIF version |
Description: Bound-variable hypothesis builder for a class abstraction. (Contributed by Mario Carneiro, 8-Oct-2016.) |
Ref | Expression |
---|---|
nfabd.1 | ⊢ Ⅎ𝑦𝜑 |
nfabd.2 | ⊢ (𝜑 → Ⅎ𝑥𝜓) |
Ref | Expression |
---|---|
nfabd | ⊢ (𝜑 → Ⅎ𝑥{𝑦 ∣ 𝜓}) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | nfv 1515 | . 2 ⊢ Ⅎ𝑧𝜑 | |
2 | df-clab 2151 | . . 3 ⊢ (𝑧 ∈ {𝑦 ∣ 𝜓} ↔ [𝑧 / 𝑦]𝜓) | |
3 | nfabd.1 | . . . 4 ⊢ Ⅎ𝑦𝜑 | |
4 | nfabd.2 | . . . 4 ⊢ (𝜑 → Ⅎ𝑥𝜓) | |
5 | 3, 4 | nfsbd 1964 | . . 3 ⊢ (𝜑 → Ⅎ𝑥[𝑧 / 𝑦]𝜓) |
6 | 2, 5 | nfxfrd 1462 | . 2 ⊢ (𝜑 → Ⅎ𝑥 𝑧 ∈ {𝑦 ∣ 𝜓}) |
7 | 1, 6 | nfcd 2301 | 1 ⊢ (𝜑 → Ⅎ𝑥{𝑦 ∣ 𝜓}) |
Colors of variables: wff set class |
Syntax hints: → wi 4 Ⅎwnf 1447 [wsb 1749 ∈ wcel 2135 {cab 2150 Ⅎwnfc 2293 |
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-io 699 ax-5 1434 ax-7 1435 ax-gen 1436 ax-ie1 1480 ax-ie2 1481 ax-8 1491 ax-10 1492 ax-11 1493 ax-i12 1494 ax-bndl 1496 ax-4 1497 ax-17 1513 ax-i9 1517 ax-ial 1521 ax-i5r 1522 |
This theorem depends on definitions: df-bi 116 df-nf 1448 df-sb 1750 df-clab 2151 df-nfc 2295 |
This theorem is referenced by: nfsbcd 2965 nfcsb1d 3071 nfcsbd 3075 nfifd 3542 nfunid 3790 nfiotadw 5150 nfixpxy 6674 |
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