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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 2375. Use the weaker nfabdw 2925 when possible. (Contributed by Mario Carneiro, 8-Oct-2016.) Avoid ax-9 2116 and ax-ext 2706. (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 1912 | . 2 ⊢ Ⅎ𝑧𝜑 | |
2 | df-clab 2713 | . . 3 ⊢ (𝑧 ∈ {𝑦 ∣ 𝜓} ↔ [𝑧 / 𝑦]𝜓) | |
3 | nfabd.1 | . . . 4 ⊢ Ⅎ𝑦𝜑 | |
4 | nfabd.2 | . . . 4 ⊢ (𝜑 → Ⅎ𝑥𝜓) | |
5 | 3, 4 | nfsbd 2525 | . . 3 ⊢ (𝜑 → Ⅎ𝑥[𝑧 / 𝑦]𝜓) |
6 | 2, 5 | nfxfrd 1851 | . 2 ⊢ (𝜑 → Ⅎ𝑥 𝑧 ∈ {𝑦 ∣ 𝜓}) |
7 | 1, 6 | nfcd 2896 | 1 ⊢ (𝜑 → Ⅎ𝑥{𝑦 ∣ 𝜓}) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 Ⅎwnf 1780 [wsb 2062 ∈ wcel 2106 {cab 2712 Ⅎwnfc 2888 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1792 ax-4 1806 ax-5 1908 ax-6 1965 ax-7 2005 ax-10 2139 ax-11 2155 ax-12 2175 ax-13 2375 |
This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-tru 1540 df-ex 1777 df-nf 1781 df-sb 2063 df-clab 2713 df-nfc 2890 |
This theorem is referenced by: nfabd2 2927 nfsbcd 3815 nfcsbd 3934 nfiotad 6521 nfiundg 48906 |
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