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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. (Contributed by Mario Carneiro, 8-Oct-2016.) Avoid ax-9 2116 and ax-ext 2754. (Revised by Wolf Lammen, 23-May-2023.) |
Ref | Expression |
---|---|
nfabd.1 | ⊢ Ⅎ𝑦𝜑 |
nfabd.2 | ⊢ (𝜑 → Ⅎ𝑥𝜓) |
Ref | Expression |
---|---|
nfabd | ⊢ (𝜑 → Ⅎ𝑥{𝑦 ∣ 𝜓}) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | nfv 1957 | . 2 ⊢ Ⅎ𝑧𝜑 | |
2 | df-clab 2764 | . . 3 ⊢ (𝑧 ∈ {𝑦 ∣ 𝜓} ↔ [𝑧 / 𝑦]𝜓) | |
3 | nfabd.1 | . . . 4 ⊢ Ⅎ𝑦𝜑 | |
4 | nfabd.2 | . . . 4 ⊢ (𝜑 → Ⅎ𝑥𝜓) | |
5 | 3, 4 | nfsbd 2522 | . . 3 ⊢ (𝜑 → Ⅎ𝑥[𝑧 / 𝑦]𝜓) |
6 | 2, 5 | nfxfrd 1898 | . 2 ⊢ (𝜑 → Ⅎ𝑥 𝑧 ∈ {𝑦 ∣ 𝜓}) |
7 | 1, 6 | nfcd 2926 | 1 ⊢ (𝜑 → Ⅎ𝑥{𝑦 ∣ 𝜓}) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 Ⅎwnf 1827 [wsb 2011 ∈ wcel 2107 {cab 2763 Ⅎwnfc 2919 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1839 ax-4 1853 ax-5 1953 ax-6 2021 ax-7 2055 ax-10 2135 ax-11 2150 ax-12 2163 ax-13 2334 |
This theorem depends on definitions: df-bi 199 df-an 387 df-or 837 df-tru 1605 df-ex 1824 df-nf 1828 df-sb 2012 df-clab 2764 df-nfc 2921 |
This theorem is referenced by: nfabd2 2955 nfsbcd 3673 nfcsb1d 3765 nfcsbd 3768 nfifd 4335 nfunid 4680 nfiotad 6104 nfintd 43539 nfiund 43540 |
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