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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 5212 | . 2 ⊢ {〈𝑥, 𝑦〉 ∣ 𝜓} = {𝑤 ∣ ∃𝑥∃𝑦(𝑤 = 〈𝑥, 𝑦〉 ∧ 𝜓)} | |
2 | nfv 1909 | . . 3 ⊢ Ⅎ𝑤𝜑 | |
3 | nfopabd.1 | . . . 4 ⊢ Ⅎ𝑥𝜑 | |
4 | nfopabd.2 | . . . . 5 ⊢ Ⅎ𝑦𝜑 | |
5 | nfvd 1910 | . . . . . 6 ⊢ (𝜑 → Ⅎ𝑧 𝑤 = 〈𝑥, 𝑦〉) | |
6 | nfopabd.4 | . . . . . 6 ⊢ (𝜑 → Ⅎ𝑧𝜓) | |
7 | 5, 6 | nfand 1892 | . . . . 5 ⊢ (𝜑 → Ⅎ𝑧(𝑤 = 〈𝑥, 𝑦〉 ∧ 𝜓)) |
8 | 4, 7 | nfexd 2317 | . . . 4 ⊢ (𝜑 → Ⅎ𝑧∃𝑦(𝑤 = 〈𝑥, 𝑦〉 ∧ 𝜓)) |
9 | 3, 8 | nfexd 2317 | . . 3 ⊢ (𝜑 → Ⅎ𝑧∃𝑥∃𝑦(𝑤 = 〈𝑥, 𝑦〉 ∧ 𝜓)) |
10 | 2, 9 | nfabdw 2915 | . 2 ⊢ (𝜑 → Ⅎ𝑧{𝑤 ∣ ∃𝑥∃𝑦(𝑤 = 〈𝑥, 𝑦〉 ∧ 𝜓)}) |
11 | 1, 10 | nfcxfrd 2890 | 1 ⊢ (𝜑 → Ⅎ𝑧{〈𝑥, 𝑦〉 ∣ 𝜓}) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 394 = wceq 1533 ∃wex 1773 Ⅎwnf 1777 {cab 2702 Ⅎwnfc 2875 〈cop 4636 {copab 5211 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1789 ax-4 1803 ax-5 1905 ax-6 1963 ax-7 2003 ax-8 2100 ax-9 2108 ax-10 2129 ax-11 2146 ax-12 2166 ax-ext 2696 |
This theorem depends on definitions: df-bi 206 df-an 395 df-or 846 df-ex 1774 df-nf 1778 df-sb 2060 df-clab 2703 df-cleq 2717 df-clel 2802 df-nfc 2877 df-opab 5212 |
This theorem is referenced by: nfopab 5218 nfttrcld 9735 |
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