Metamath Proof Explorer |
< Previous
Next >
Nearby theorems |
||
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 5113 | . 2 ⊢ {〈𝑥, 𝑦〉 ∣ 𝜓} = {𝑤 ∣ ∃𝑥∃𝑦(𝑤 = 〈𝑥, 𝑦〉 ∧ 𝜓)} | |
2 | nfv 1922 | . . 3 ⊢ Ⅎ𝑤𝜑 | |
3 | nfopabd.1 | . . . 4 ⊢ Ⅎ𝑥𝜑 | |
4 | nfopabd.2 | . . . . 5 ⊢ Ⅎ𝑦𝜑 | |
5 | nfvd 1923 | . . . . . 6 ⊢ (𝜑 → Ⅎ𝑧 𝑤 = 〈𝑥, 𝑦〉) | |
6 | nfopabd.4 | . . . . . 6 ⊢ (𝜑 → Ⅎ𝑧𝜓) | |
7 | 5, 6 | nfand 1905 | . . . . 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 2927 | . 2 ⊢ (𝜑 → Ⅎ𝑧{𝑤 ∣ ∃𝑥∃𝑦(𝑤 = 〈𝑥, 𝑦〉 ∧ 𝜓)}) |
11 | 1, 10 | nfcxfrd 2903 | 1 ⊢ (𝜑 → Ⅎ𝑧{〈𝑥, 𝑦〉 ∣ 𝜓}) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 399 = wceq 1543 ∃wex 1787 Ⅎwnf 1791 {cab 2714 Ⅎwnfc 2884 〈cop 4544 {copab 5112 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1803 ax-4 1817 ax-5 1918 ax-6 1976 ax-7 2016 ax-8 2112 ax-9 2120 ax-10 2141 ax-11 2158 ax-12 2175 ax-ext 2708 |
This theorem depends on definitions: df-bi 210 df-an 400 df-or 848 df-ex 1788 df-nf 1792 df-sb 2071 df-clab 2715 df-cleq 2729 df-clel 2816 df-nfc 2886 df-opab 5113 |
This theorem is referenced by: nfopab 5119 nfttrcld 33506 |
Copyright terms: Public domain | W3C validator |