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Mirrors > Home > ILE Home > Th. List > preq12i | GIF version |
Description: Equality inference for unordered pairs. (Contributed by NM, 19-Oct-2012.) |
Ref | Expression |
---|---|
preq1i.1 | ⊢ 𝐴 = 𝐵 |
preq12i.2 | ⊢ 𝐶 = 𝐷 |
Ref | Expression |
---|---|
preq12i | ⊢ {𝐴, 𝐶} = {𝐵, 𝐷} |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | preq1i.1 | . 2 ⊢ 𝐴 = 𝐵 | |
2 | preq12i.2 | . 2 ⊢ 𝐶 = 𝐷 | |
3 | preq12 3495 | . 2 ⊢ ((𝐴 = 𝐵 ∧ 𝐶 = 𝐷) → {𝐴, 𝐶} = {𝐵, 𝐷}) | |
4 | 1, 2, 3 | mp2an 417 | 1 ⊢ {𝐴, 𝐶} = {𝐵, 𝐷} |
Colors of variables: wff set class |
Syntax hints: = wceq 1285 {cpr 3423 |
This theorem was proved from axioms: ax-1 5 ax-2 6 ax-mp 7 ax-ia1 104 ax-ia2 105 ax-ia3 106 ax-io 663 ax-5 1377 ax-7 1378 ax-gen 1379 ax-ie1 1423 ax-ie2 1424 ax-8 1436 ax-10 1437 ax-11 1438 ax-i12 1439 ax-bndl 1440 ax-4 1441 ax-17 1460 ax-i9 1464 ax-ial 1468 ax-i5r 1469 ax-ext 2065 |
This theorem depends on definitions: df-bi 115 df-tru 1288 df-nf 1391 df-sb 1688 df-clab 2070 df-cleq 2076 df-clel 2079 df-nfc 2212 df-v 2614 df-un 2988 df-sn 3428 df-pr 3429 |
This theorem is referenced by: (None) |
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