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Mirrors > Home > MPE Home > Th. List > nfoprab3 | Structured version Visualization version GIF version |
Description: The abstraction variables in an operation class abstraction are not free. (Contributed by NM, 22-Aug-2013.) |
Ref | Expression |
---|---|
nfoprab3 | ⊢ Ⅎ𝑧{〈〈𝑥, 𝑦〉, 𝑧〉 ∣ 𝜑} |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | df-oprab 7162 | . 2 ⊢ {〈〈𝑥, 𝑦〉, 𝑧〉 ∣ 𝜑} = {𝑤 ∣ ∃𝑥∃𝑦∃𝑧(𝑤 = 〈〈𝑥, 𝑦〉, 𝑧〉 ∧ 𝜑)} | |
2 | nfe1 2154 | . . . . 5 ⊢ Ⅎ𝑧∃𝑧(𝑤 = 〈〈𝑥, 𝑦〉, 𝑧〉 ∧ 𝜑) | |
3 | 2 | nfex 2343 | . . . 4 ⊢ Ⅎ𝑧∃𝑦∃𝑧(𝑤 = 〈〈𝑥, 𝑦〉, 𝑧〉 ∧ 𝜑) |
4 | 3 | nfex 2343 | . . 3 ⊢ Ⅎ𝑧∃𝑥∃𝑦∃𝑧(𝑤 = 〈〈𝑥, 𝑦〉, 𝑧〉 ∧ 𝜑) |
5 | 4 | nfab 2986 | . 2 ⊢ Ⅎ𝑧{𝑤 ∣ ∃𝑥∃𝑦∃𝑧(𝑤 = 〈〈𝑥, 𝑦〉, 𝑧〉 ∧ 𝜑)} |
6 | 1, 5 | nfcxfr 2977 | 1 ⊢ Ⅎ𝑧{〈〈𝑥, 𝑦〉, 𝑧〉 ∣ 𝜑} |
Colors of variables: wff setvar class |
Syntax hints: ∧ wa 398 = wceq 1537 ∃wex 1780 {cab 2801 Ⅎwnfc 2963 〈cop 4575 {coprab 7159 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1796 ax-4 1810 ax-5 1911 ax-6 1970 ax-7 2015 ax-8 2116 ax-9 2124 ax-10 2145 ax-11 2161 ax-12 2177 ax-ext 2795 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-ex 1781 df-nf 1785 df-sb 2070 df-clab 2802 df-cleq 2816 df-clel 2895 df-nfc 2965 df-oprab 7162 |
This theorem is referenced by: ssoprab2b 7225 eqoprab2bw 7226 ov3 7313 tposoprab 7930 |
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