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Mirrors > Home > MPE Home > Th. List > nfoprab | Structured version Visualization version GIF version |
Description: Bound-variable hypothesis builder for an operation class abstraction. (Contributed by NM, 22-Aug-2013.) |
Ref | Expression |
---|---|
nfoprab.1 | ⊢ Ⅎ𝑤𝜑 |
Ref | Expression |
---|---|
nfoprab | ⊢ Ⅎ𝑤{〈〈𝑥, 𝑦〉, 𝑧〉 ∣ 𝜑} |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | df-oprab 6654 | . 2 ⊢ {〈〈𝑥, 𝑦〉, 𝑧〉 ∣ 𝜑} = {𝑣 ∣ ∃𝑥∃𝑦∃𝑧(𝑣 = 〈〈𝑥, 𝑦〉, 𝑧〉 ∧ 𝜑)} | |
2 | nfv 1843 | . . . . . . 7 ⊢ Ⅎ𝑤 𝑣 = 〈〈𝑥, 𝑦〉, 𝑧〉 | |
3 | nfoprab.1 | . . . . . . 7 ⊢ Ⅎ𝑤𝜑 | |
4 | 2, 3 | nfan 1828 | . . . . . 6 ⊢ Ⅎ𝑤(𝑣 = 〈〈𝑥, 𝑦〉, 𝑧〉 ∧ 𝜑) |
5 | 4 | nfex 2154 | . . . . 5 ⊢ Ⅎ𝑤∃𝑧(𝑣 = 〈〈𝑥, 𝑦〉, 𝑧〉 ∧ 𝜑) |
6 | 5 | nfex 2154 | . . . 4 ⊢ Ⅎ𝑤∃𝑦∃𝑧(𝑣 = 〈〈𝑥, 𝑦〉, 𝑧〉 ∧ 𝜑) |
7 | 6 | nfex 2154 | . . 3 ⊢ Ⅎ𝑤∃𝑥∃𝑦∃𝑧(𝑣 = 〈〈𝑥, 𝑦〉, 𝑧〉 ∧ 𝜑) |
8 | 7 | nfab 2769 | . 2 ⊢ Ⅎ𝑤{𝑣 ∣ ∃𝑥∃𝑦∃𝑧(𝑣 = 〈〈𝑥, 𝑦〉, 𝑧〉 ∧ 𝜑)} |
9 | 1, 8 | nfcxfr 2762 | 1 ⊢ Ⅎ𝑤{〈〈𝑥, 𝑦〉, 𝑧〉 ∣ 𝜑} |
Colors of variables: wff setvar class |
Syntax hints: ∧ wa 384 = wceq 1483 ∃wex 1704 Ⅎwnf 1708 {cab 2608 Ⅎwnfc 2751 〈cop 4183 {coprab 6651 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1722 ax-4 1737 ax-5 1839 ax-6 1888 ax-7 1935 ax-9 1999 ax-10 2019 ax-11 2034 ax-12 2047 ax-13 2246 ax-ext 2602 |
This theorem depends on definitions: df-bi 197 df-or 385 df-an 386 df-tru 1486 df-ex 1705 df-nf 1710 df-sb 1881 df-clab 2609 df-cleq 2615 df-clel 2618 df-nfc 2753 df-oprab 6654 |
This theorem is referenced by: nfmpt2 6724 |
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