Metamath Proof Explorer |
< Previous
Next >
Nearby theorems |
||
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 7195 | . 2 ⊢ {〈〈𝑥, 𝑦〉, 𝑧〉 ∣ 𝜑} = {𝑤 ∣ ∃𝑥∃𝑦∃𝑧(𝑤 = 〈〈𝑥, 𝑦〉, 𝑧〉 ∧ 𝜑)} | |
2 | nfe1 2153 | . . . . 5 ⊢ Ⅎ𝑧∃𝑧(𝑤 = 〈〈𝑥, 𝑦〉, 𝑧〉 ∧ 𝜑) | |
3 | 2 | nfex 2325 | . . . 4 ⊢ Ⅎ𝑧∃𝑦∃𝑧(𝑤 = 〈〈𝑥, 𝑦〉, 𝑧〉 ∧ 𝜑) |
4 | 3 | nfex 2325 | . . 3 ⊢ Ⅎ𝑧∃𝑥∃𝑦∃𝑧(𝑤 = 〈〈𝑥, 𝑦〉, 𝑧〉 ∧ 𝜑) |
5 | 4 | nfab 2903 | . 2 ⊢ Ⅎ𝑧{𝑤 ∣ ∃𝑥∃𝑦∃𝑧(𝑤 = 〈〈𝑥, 𝑦〉, 𝑧〉 ∧ 𝜑)} |
6 | 1, 5 | nfcxfr 2895 | 1 ⊢ Ⅎ𝑧{〈〈𝑥, 𝑦〉, 𝑧〉 ∣ 𝜑} |
Colors of variables: wff setvar class |
Syntax hints: ∧ wa 399 = wceq 1543 ∃wex 1787 {cab 2714 Ⅎwnfc 2877 〈cop 4533 {coprab 7192 |
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 2018 ax-8 2114 ax-9 2122 ax-10 2143 ax-11 2160 ax-12 2177 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 2073 df-clab 2715 df-cleq 2728 df-clel 2809 df-nfc 2879 df-oprab 7195 |
This theorem is referenced by: ssoprab2b 7258 eqoprab2bw 7259 ov3 7349 tposoprab 7982 |
Copyright terms: Public domain | W3C validator |