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| Mirrors > Home > MPE Home > Th. List > nfabd | Structured version Visualization version GIF version | ||
| Description: Bound-variable hypothesis builder for a class abstraction. Usage of this theorem is discouraged because it depends on ax-13 2377. Use the weaker nfabdw 2927 when possible. (Contributed by Mario Carneiro, 8-Oct-2016.) Avoid ax-9 2118 and ax-ext 2708. (Revised by Wolf Lammen, 23-May-2023.) (New usage is discouraged.) |
| Ref | Expression |
|---|---|
| nfabd.1 | ⊢ Ⅎ𝑦𝜑 |
| nfabd.2 | ⊢ (𝜑 → Ⅎ𝑥𝜓) |
| Ref | Expression |
|---|---|
| nfabd | ⊢ (𝜑 → Ⅎ𝑥{𝑦 ∣ 𝜓}) |
| Step | Hyp | Ref | Expression |
|---|---|---|---|
| 1 | nfv 1914 | . 2 ⊢ Ⅎ𝑧𝜑 | |
| 2 | df-clab 2715 | . . 3 ⊢ (𝑧 ∈ {𝑦 ∣ 𝜓} ↔ [𝑧 / 𝑦]𝜓) | |
| 3 | nfabd.1 | . . . 4 ⊢ Ⅎ𝑦𝜑 | |
| 4 | nfabd.2 | . . . 4 ⊢ (𝜑 → Ⅎ𝑥𝜓) | |
| 5 | 3, 4 | nfsbd 2527 | . . 3 ⊢ (𝜑 → Ⅎ𝑥[𝑧 / 𝑦]𝜓) |
| 6 | 2, 5 | nfxfrd 1854 | . 2 ⊢ (𝜑 → Ⅎ𝑥 𝑧 ∈ {𝑦 ∣ 𝜓}) |
| 7 | 1, 6 | nfcd 2898 | 1 ⊢ (𝜑 → Ⅎ𝑥{𝑦 ∣ 𝜓}) |
| Colors of variables: wff setvar class |
| Syntax hints: → wi 4 Ⅎwnf 1783 [wsb 2064 ∈ wcel 2108 {cab 2714 Ⅎwnfc 2890 |
| 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-10 2141 ax-11 2157 ax-12 2177 ax-13 2377 |
| This theorem depends on definitions: df-bi 207 df-an 396 df-or 849 df-tru 1543 df-ex 1780 df-nf 1784 df-sb 2065 df-clab 2715 df-nfc 2892 |
| This theorem is referenced by: nfabd2 2929 nfsbcd 3812 nfcsbd 3924 nfiotad 6519 nfiundg 49194 |
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