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Mirrors > Home > ILE Home > Th. List > nfreuxy | GIF version |
Description: Not-free for restricted uniqueness. This is a version where 𝑥 and 𝑦 are distinct. (Contributed by Jim Kingdon, 6-Jun-2018.) |
Ref | Expression |
---|---|
nfreuxy.1 | ⊢ Ⅎ𝑥𝐴 |
nfreuxy.2 | ⊢ Ⅎ𝑥𝜑 |
Ref | Expression |
---|---|
nfreuxy | ⊢ Ⅎ𝑥∃!𝑦 ∈ 𝐴 𝜑 |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | nftru 1427 | . . 3 ⊢ Ⅎ𝑦⊤ | |
2 | nfreuxy.1 | . . . 4 ⊢ Ⅎ𝑥𝐴 | |
3 | 2 | a1i 9 | . . 3 ⊢ (⊤ → Ⅎ𝑥𝐴) |
4 | nfreuxy.2 | . . . 4 ⊢ Ⅎ𝑥𝜑 | |
5 | 4 | a1i 9 | . . 3 ⊢ (⊤ → Ⅎ𝑥𝜑) |
6 | 1, 3, 5 | nfreudxy 2581 | . 2 ⊢ (⊤ → Ⅎ𝑥∃!𝑦 ∈ 𝐴 𝜑) |
7 | 6 | mptru 1325 | 1 ⊢ Ⅎ𝑥∃!𝑦 ∈ 𝐴 𝜑 |
Colors of variables: wff set class |
Syntax hints: ⊤wtru 1317 Ⅎwnf 1421 Ⅎwnfc 2245 ∃!wreu 2395 |
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 683 ax-5 1408 ax-7 1409 ax-gen 1410 ax-ie1 1454 ax-ie2 1455 ax-8 1467 ax-10 1468 ax-11 1469 ax-i12 1470 ax-bndl 1471 ax-4 1472 ax-17 1491 ax-i9 1495 ax-ial 1499 ax-i5r 1500 ax-ext 2099 |
This theorem depends on definitions: df-bi 116 df-tru 1319 df-nf 1422 df-sb 1721 df-eu 1980 df-cleq 2110 df-clel 2113 df-nfc 2247 df-reu 2400 |
This theorem is referenced by: sbcreug 2961 |
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