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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 1528 | . 2 ⊢ Ⅎ𝑧𝜑 | |
2 | df-clab 2164 | . . 3 ⊢ (𝑧 ∈ {𝑦 ∣ 𝜓} ↔ [𝑧 / 𝑦]𝜓) | |
3 | nfabd.1 | . . . 4 ⊢ Ⅎ𝑦𝜑 | |
4 | nfabd.2 | . . . 4 ⊢ (𝜑 → Ⅎ𝑥𝜓) | |
5 | 3, 4 | nfsbd 1977 | . . 3 ⊢ (𝜑 → Ⅎ𝑥[𝑧 / 𝑦]𝜓) |
6 | 2, 5 | nfxfrd 1475 | . 2 ⊢ (𝜑 → Ⅎ𝑥 𝑧 ∈ {𝑦 ∣ 𝜓}) |
7 | 1, 6 | nfcd 2314 | 1 ⊢ (𝜑 → Ⅎ𝑥{𝑦 ∣ 𝜓}) |
Colors of variables: wff set class |
Syntax hints: → wi 4 Ⅎwnf 1460 [wsb 1762 ∈ wcel 2148 {cab 2163 Ⅎwnfc 2306 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-ia1 106 ax-ia2 107 ax-ia3 108 ax-io 709 ax-5 1447 ax-7 1448 ax-gen 1449 ax-ie1 1493 ax-ie2 1494 ax-8 1504 ax-10 1505 ax-11 1506 ax-i12 1507 ax-bndl 1509 ax-4 1510 ax-17 1526 ax-i9 1530 ax-ial 1534 ax-i5r 1535 |
This theorem depends on definitions: df-bi 117 df-nf 1461 df-sb 1763 df-clab 2164 df-nfc 2308 |
This theorem is referenced by: nfsbcd 2982 nfcsb1d 3088 nfcsbd 3092 nfifd 3561 nfunid 3814 nfiotadw 5177 nfixpxy 6711 |
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