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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 5211 | . 2 ⊢ {〈𝑥, 𝑦〉 ∣ 𝜓} = {𝑤 ∣ ∃𝑥∃𝑦(𝑤 = 〈𝑥, 𝑦〉 ∧ 𝜓)} | |
2 | nfv 1912 | . . 3 ⊢ Ⅎ𝑤𝜑 | |
3 | nfopabd.1 | . . . 4 ⊢ Ⅎ𝑥𝜑 | |
4 | nfopabd.2 | . . . . 5 ⊢ Ⅎ𝑦𝜑 | |
5 | nfvd 1913 | . . . . . 6 ⊢ (𝜑 → Ⅎ𝑧 𝑤 = 〈𝑥, 𝑦〉) | |
6 | nfopabd.4 | . . . . . 6 ⊢ (𝜑 → Ⅎ𝑧𝜓) | |
7 | 5, 6 | nfand 1895 | . . . . 5 ⊢ (𝜑 → Ⅎ𝑧(𝑤 = 〈𝑥, 𝑦〉 ∧ 𝜓)) |
8 | 4, 7 | nfexd 2328 | . . . 4 ⊢ (𝜑 → Ⅎ𝑧∃𝑦(𝑤 = 〈𝑥, 𝑦〉 ∧ 𝜓)) |
9 | 3, 8 | nfexd 2328 | . . 3 ⊢ (𝜑 → Ⅎ𝑧∃𝑥∃𝑦(𝑤 = 〈𝑥, 𝑦〉 ∧ 𝜓)) |
10 | 2, 9 | nfabdw 2925 | . 2 ⊢ (𝜑 → Ⅎ𝑧{𝑤 ∣ ∃𝑥∃𝑦(𝑤 = 〈𝑥, 𝑦〉 ∧ 𝜓)}) |
11 | 1, 10 | nfcxfrd 2902 | 1 ⊢ (𝜑 → Ⅎ𝑧{〈𝑥, 𝑦〉 ∣ 𝜓}) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 395 = wceq 1537 ∃wex 1776 Ⅎwnf 1780 {cab 2712 Ⅎwnfc 2888 〈cop 4637 {copab 5210 |
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-8 2108 ax-9 2116 ax-10 2139 ax-11 2155 ax-12 2175 ax-ext 2706 |
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-cleq 2727 df-clel 2814 df-nfc 2890 df-opab 5211 |
This theorem is referenced by: nfopab 5217 nfttrcld 9748 |
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