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Mirrors > Home > MPE Home > Th. List > nfopab | Structured version Visualization version GIF version |
Description: Bound-variable hypothesis builder for class abstraction. (Contributed by NM, 1-Sep-1999.) (Unnecessary distinct variable restrictions were removed by Andrew Salmon, 11-Jul-2011.) |
Ref | Expression |
---|---|
nfopab.1 | ⊢ Ⅎ𝑧𝜑 |
Ref | Expression |
---|---|
nfopab | ⊢ Ⅎ𝑧{〈𝑥, 𝑦〉 ∣ 𝜑} |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | df-opab 4644 | . 2 ⊢ {〈𝑥, 𝑦〉 ∣ 𝜑} = {𝑤 ∣ ∃𝑥∃𝑦(𝑤 = 〈𝑥, 𝑦〉 ∧ 𝜑)} | |
2 | nfv 1830 | . . . . . 6 ⊢ Ⅎ𝑧 𝑤 = 〈𝑥, 𝑦〉 | |
3 | nfopab.1 | . . . . . 6 ⊢ Ⅎ𝑧𝜑 | |
4 | 2, 3 | nfan 1816 | . . . . 5 ⊢ Ⅎ𝑧(𝑤 = 〈𝑥, 𝑦〉 ∧ 𝜑) |
5 | 4 | nfex 2140 | . . . 4 ⊢ Ⅎ𝑧∃𝑦(𝑤 = 〈𝑥, 𝑦〉 ∧ 𝜑) |
6 | 5 | nfex 2140 | . . 3 ⊢ Ⅎ𝑧∃𝑥∃𝑦(𝑤 = 〈𝑥, 𝑦〉 ∧ 𝜑) |
7 | 6 | nfab 2755 | . 2 ⊢ Ⅎ𝑧{𝑤 ∣ ∃𝑥∃𝑦(𝑤 = 〈𝑥, 𝑦〉 ∧ 𝜑)} |
8 | 1, 7 | nfcxfr 2749 | 1 ⊢ Ⅎ𝑧{〈𝑥, 𝑦〉 ∣ 𝜑} |
Colors of variables: wff setvar class |
Syntax hints: ∧ wa 383 = wceq 1475 ∃wex 1695 Ⅎwnf 1699 {cab 2596 Ⅎwnfc 2738 〈cop 4131 {copab 4642 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1713 ax-4 1728 ax-5 1827 ax-6 1875 ax-7 1922 ax-10 2006 ax-11 2021 ax-12 2034 ax-13 2234 ax-ext 2590 |
This theorem depends on definitions: df-bi 196 df-or 384 df-an 385 df-tru 1478 df-ex 1696 df-nf 1701 df-sb 1868 df-clab 2597 df-cleq 2603 df-clel 2606 df-nfc 2740 df-opab 4644 |
This theorem is referenced by: nfmpt 4674 csbopab 4932 csbopabgALT 4933 nfxp 5066 nfco 5209 nfcnv 5223 |
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