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Mirrors > Home > ILE Home > Th. List > opeq12i | GIF version |
Description: Equality inference for ordered pairs. (Contributed by NM, 16-Dec-2006.) (Proof shortened by Eric Schmidt, 4-Apr-2007.) |
Ref | Expression |
---|---|
opeq1i.1 | ⊢ 𝐴 = 𝐵 |
opeq12i.2 | ⊢ 𝐶 = 𝐷 |
Ref | Expression |
---|---|
opeq12i | ⊢ 〈𝐴, 𝐶〉 = 〈𝐵, 𝐷〉 |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | opeq1i.1 | . 2 ⊢ 𝐴 = 𝐵 | |
2 | opeq12i.2 | . 2 ⊢ 𝐶 = 𝐷 | |
3 | opeq12 3707 | . 2 ⊢ ((𝐴 = 𝐵 ∧ 𝐶 = 𝐷) → 〈𝐴, 𝐶〉 = 〈𝐵, 𝐷〉) | |
4 | 1, 2, 3 | mp2an 422 | 1 ⊢ 〈𝐴, 𝐶〉 = 〈𝐵, 𝐷〉 |
Colors of variables: wff set class |
Syntax hints: = wceq 1331 〈cop 3530 |
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-3an 964 df-tru 1334 df-nf 1437 df-sb 1736 df-clab 2126 df-cleq 2132 df-clel 2135 df-nfc 2270 df-v 2688 df-un 3075 df-sn 3533 df-pr 3534 df-op 3536 |
This theorem is referenced by: addpinq1 7272 genipv 7317 ltexpri 7421 recexpr 7446 cauappcvgprlemladdru 7464 cauappcvgprlemladdrl 7465 cauappcvgpr 7470 caucvgprlemcl 7484 caucvgprlemladdrl 7486 caucvgpr 7490 caucvgprprlemval 7496 caucvgprprlemnbj 7501 caucvgprprlemmu 7503 caucvgprprlemclphr 7513 caucvgprprlemaddq 7516 caucvgprprlem1 7517 caucvgprprlem2 7518 caucvgsr 7610 pitonnlem1 7653 axi2m1 7683 axcaucvg 7708 |
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