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| Mirrors > Home > MPE Home > Th. List > nfopabd | Structured version Visualization version GIF version | ||
| Description: Bound-variable hypothesis builder for class abstraction. Deduction form. (Contributed by Scott Fenton, 26-Oct-2024.) | 
| Ref | Expression | 
|---|---|
| nfopabd.1 | ⊢ Ⅎ𝑥𝜑 | 
| nfopabd.2 | ⊢ Ⅎ𝑦𝜑 | 
| nfopabd.4 | ⊢ (𝜑 → Ⅎ𝑧𝜓) | 
| Ref | Expression | 
|---|---|
| nfopabd | ⊢ (𝜑 → Ⅎ𝑧{〈𝑥, 𝑦〉 ∣ 𝜓}) | 
| Step | Hyp | Ref | Expression | 
|---|---|---|---|
| 1 | df-opab 5206 | . 2 ⊢ {〈𝑥, 𝑦〉 ∣ 𝜓} = {𝑤 ∣ ∃𝑥∃𝑦(𝑤 = 〈𝑥, 𝑦〉 ∧ 𝜓)} | |
| 2 | nfv 1914 | . . 3 ⊢ Ⅎ𝑤𝜑 | |
| 3 | nfopabd.1 | . . . 4 ⊢ Ⅎ𝑥𝜑 | |
| 4 | nfopabd.2 | . . . . 5 ⊢ Ⅎ𝑦𝜑 | |
| 5 | nfvd 1915 | . . . . . 6 ⊢ (𝜑 → Ⅎ𝑧 𝑤 = 〈𝑥, 𝑦〉) | |
| 6 | nfopabd.4 | . . . . . 6 ⊢ (𝜑 → Ⅎ𝑧𝜓) | |
| 7 | 5, 6 | nfand 1897 | . . . . 5 ⊢ (𝜑 → Ⅎ𝑧(𝑤 = 〈𝑥, 𝑦〉 ∧ 𝜓)) | 
| 8 | 4, 7 | nfexd 2329 | . . . 4 ⊢ (𝜑 → Ⅎ𝑧∃𝑦(𝑤 = 〈𝑥, 𝑦〉 ∧ 𝜓)) | 
| 9 | 3, 8 | nfexd 2329 | . . 3 ⊢ (𝜑 → Ⅎ𝑧∃𝑥∃𝑦(𝑤 = 〈𝑥, 𝑦〉 ∧ 𝜓)) | 
| 10 | 2, 9 | nfabdw 2927 | . 2 ⊢ (𝜑 → Ⅎ𝑧{𝑤 ∣ ∃𝑥∃𝑦(𝑤 = 〈𝑥, 𝑦〉 ∧ 𝜓)}) | 
| 11 | 1, 10 | nfcxfrd 2904 | 1 ⊢ (𝜑 → Ⅎ𝑧{〈𝑥, 𝑦〉 ∣ 𝜓}) | 
| Colors of variables: wff setvar class | 
| Syntax hints: → wi 4 ∧ wa 395 = wceq 1540 ∃wex 1779 Ⅎwnf 1783 {cab 2714 Ⅎwnfc 2890 〈cop 4632 {copab 5205 | 
| 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-8 2110 ax-9 2118 ax-10 2141 ax-11 2157 ax-12 2177 ax-ext 2708 | 
| This theorem depends on definitions: df-bi 207 df-an 396 df-or 849 df-ex 1780 df-nf 1784 df-sb 2065 df-clab 2715 df-cleq 2729 df-clel 2816 df-nfc 2892 df-opab 5206 | 
| This theorem is referenced by: nfopab 5212 nfttrcld 9750 | 
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