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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 2924 with a disjoint variable condition, which does not require ax-13 2372. (Contributed by Mario Carneiro, 8-Oct-2016.) (Revised by Gino Giotto, 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 1921 | . 2 ⊢ Ⅎ𝑧𝜑 | |
2 | df-clab 2717 | . . . 4 ⊢ (𝑧 ∈ {𝑦 ∣ 𝜓} ↔ [𝑧 / 𝑦]𝜓) | |
3 | sb6 2095 | . . . 4 ⊢ ([𝑧 / 𝑦]𝜓 ↔ ∀𝑦(𝑦 = 𝑧 → 𝜓)) | |
4 | 2, 3 | bitri 278 | . . 3 ⊢ (𝑧 ∈ {𝑦 ∣ 𝜓} ↔ ∀𝑦(𝑦 = 𝑧 → 𝜓)) |
5 | nfabdw.1 | . . . 4 ⊢ Ⅎ𝑦𝜑 | |
6 | nfvd 1922 | . . . . 5 ⊢ (𝜑 → Ⅎ𝑥 𝑦 = 𝑧) | |
7 | nfabdw.2 | . . . . 5 ⊢ (𝜑 → Ⅎ𝑥𝜓) | |
8 | 6, 7 | nfimd 1901 | . . . 4 ⊢ (𝜑 → Ⅎ𝑥(𝑦 = 𝑧 → 𝜓)) |
9 | 5, 8 | nfald 2330 | . . 3 ⊢ (𝜑 → Ⅎ𝑥∀𝑦(𝑦 = 𝑧 → 𝜓)) |
10 | 4, 9 | nfxfrd 1860 | . 2 ⊢ (𝜑 → Ⅎ𝑥 𝑧 ∈ {𝑦 ∣ 𝜓}) |
11 | 1, 10 | nfcd 2887 | 1 ⊢ (𝜑 → Ⅎ𝑥{𝑦 ∣ 𝜓}) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∀wal 1540 Ⅎwnf 1790 [wsb 2074 ∈ wcel 2114 {cab 2716 Ⅎwnfc 2879 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1802 ax-4 1816 ax-5 1917 ax-6 1975 ax-7 2020 ax-10 2145 ax-11 2162 ax-12 2179 |
This theorem depends on definitions: df-bi 210 df-an 400 df-or 847 df-ex 1787 df-nf 1791 df-sb 2075 df-clab 2717 df-nfc 2881 |
This theorem is referenced by: nfrabw 3288 nfsbcdw 3701 nfcsb1d 3812 nfcsbw 3816 nfifd 4443 nfunid 4802 nfiotadw 6300 nfintd 45852 nfiund 45853 |
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