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Mirrors > Home > MPE Home > Th. List > nfabdw | Structured version Visualization version GIF version |
Description: Bound-variable hypothesis builder for a class abstraction. Version of nfabd 2926 with a disjoint variable condition, which does not require ax-13 2375. (Contributed by Mario Carneiro, 8-Oct-2016.) Avoid ax-13 2375. (Revised by GG, 10-Jan-2024.) (Proof shortened by Wolf Lammen, 23-Sep-2024.) |
Ref | Expression |
---|---|
nfabdw.1 | ⊢ Ⅎ𝑦𝜑 |
nfabdw.2 | ⊢ (𝜑 → Ⅎ𝑥𝜓) |
Ref | Expression |
---|---|
nfabdw | ⊢ (𝜑 → Ⅎ𝑥{𝑦 ∣ 𝜓}) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | nfv 1912 | . 2 ⊢ Ⅎ𝑧𝜑 | |
2 | df-clab 2713 | . . . 4 ⊢ (𝑧 ∈ {𝑦 ∣ 𝜓} ↔ [𝑧 / 𝑦]𝜓) | |
3 | sb6 2083 | . . . 4 ⊢ ([𝑧 / 𝑦]𝜓 ↔ ∀𝑦(𝑦 = 𝑧 → 𝜓)) | |
4 | 2, 3 | bitri 275 | . . 3 ⊢ (𝑧 ∈ {𝑦 ∣ 𝜓} ↔ ∀𝑦(𝑦 = 𝑧 → 𝜓)) |
5 | nfabdw.1 | . . . 4 ⊢ Ⅎ𝑦𝜑 | |
6 | nfvd 1913 | . . . . 5 ⊢ (𝜑 → Ⅎ𝑥 𝑦 = 𝑧) | |
7 | nfabdw.2 | . . . . 5 ⊢ (𝜑 → Ⅎ𝑥𝜓) | |
8 | 6, 7 | nfimd 1892 | . . . 4 ⊢ (𝜑 → Ⅎ𝑥(𝑦 = 𝑧 → 𝜓)) |
9 | 5, 8 | nfald 2327 | . . 3 ⊢ (𝜑 → Ⅎ𝑥∀𝑦(𝑦 = 𝑧 → 𝜓)) |
10 | 4, 9 | nfxfrd 1851 | . 2 ⊢ (𝜑 → Ⅎ𝑥 𝑧 ∈ {𝑦 ∣ 𝜓}) |
11 | 1, 10 | nfcd 2896 | 1 ⊢ (𝜑 → Ⅎ𝑥{𝑦 ∣ 𝜓}) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∀wal 1535 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 |
This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-ex 1777 df-nf 1781 df-sb 2063 df-clab 2713 df-nfc 2890 |
This theorem is referenced by: nfrabw 3473 nfrabwOLD 3474 nfsbcdw 3812 nfcsb1d 3931 nfcsbw 3935 nfifd 4560 nfunid 4918 nfopabd 5216 nfiotadw 6519 nfintd 48904 nfiund 48905 |
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