Intuitionistic Logic Explorer |
< Previous
Next >
Nearby theorems |
||
Mirrors > Home > ILE Home > Th. List > nfopab2 | GIF version |
Description: The second abstraction variable in an ordered-pair class abstraction (class builder) is effectively not free. (Contributed by NM, 16-May-1995.) (Revised by Mario Carneiro, 14-Oct-2016.) |
Ref | Expression |
---|---|
nfopab2 | ⊢ Ⅎ𝑦{〈𝑥, 𝑦〉 ∣ 𝜑} |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | df-opab 3990 | . 2 ⊢ {〈𝑥, 𝑦〉 ∣ 𝜑} = {𝑧 ∣ ∃𝑥∃𝑦(𝑧 = 〈𝑥, 𝑦〉 ∧ 𝜑)} | |
2 | nfe1 1472 | . . . 4 ⊢ Ⅎ𝑦∃𝑦(𝑧 = 〈𝑥, 𝑦〉 ∧ 𝜑) | |
3 | 2 | nfex 1616 | . . 3 ⊢ Ⅎ𝑦∃𝑥∃𝑦(𝑧 = 〈𝑥, 𝑦〉 ∧ 𝜑) |
4 | 3 | nfab 2286 | . 2 ⊢ Ⅎ𝑦{𝑧 ∣ ∃𝑥∃𝑦(𝑧 = 〈𝑥, 𝑦〉 ∧ 𝜑)} |
5 | 1, 4 | nfcxfr 2278 | 1 ⊢ Ⅎ𝑦{〈𝑥, 𝑦〉 ∣ 𝜑} |
Colors of variables: wff set class |
Syntax hints: ∧ wa 103 = wceq 1331 ∃wex 1468 {cab 2125 Ⅎwnfc 2268 〈cop 3530 {copab 3988 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-ia1 105 ax-ia2 106 ax-ia3 107 ax-io 698 ax-5 1423 ax-7 1424 ax-gen 1425 ax-ie1 1469 ax-ie2 1470 ax-8 1482 ax-10 1483 ax-11 1484 ax-i12 1485 ax-bndl 1486 ax-4 1487 ax-17 1506 ax-i9 1510 ax-ial 1514 ax-i5r 1515 ax-ext 2121 |
This theorem depends on definitions: df-bi 116 df-nf 1437 df-sb 1736 df-clab 2126 df-cleq 2132 df-clel 2135 df-nfc 2270 df-opab 3990 |
This theorem is referenced by: opelopabsb 4182 ssopab2b 4198 dmopab 4750 rnopab 4786 funopab 5158 0neqopab 5816 |
Copyright terms: Public domain | W3C validator |