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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 3780 | . 2 ⊢ ((𝐴 = 𝐵 ∧ 𝐶 = 𝐷) → ⟨𝐴, 𝐶⟩ = ⟨𝐵, 𝐷⟩) | |
4 | 1, 2, 3 | mp2an 426 | 1 ⊢ ⟨𝐴, 𝐶⟩ = ⟨𝐵, 𝐷⟩ |
Colors of variables: wff set class |
Syntax hints: = wceq 1353 ⟨cop 3595 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-ia1 106 ax-ia2 107 ax-ia3 108 ax-io 709 ax-5 1447 ax-7 1448 ax-gen 1449 ax-ie1 1493 ax-ie2 1494 ax-8 1504 ax-10 1505 ax-11 1506 ax-i12 1507 ax-bndl 1509 ax-4 1510 ax-17 1526 ax-i9 1530 ax-ial 1534 ax-i5r 1535 ax-ext 2159 |
This theorem depends on definitions: df-bi 117 df-3an 980 df-tru 1356 df-nf 1461 df-sb 1763 df-clab 2164 df-cleq 2170 df-clel 2173 df-nfc 2308 df-v 2739 df-un 3133 df-sn 3598 df-pr 3599 df-op 3601 |
This theorem is referenced by: addpinq1 7462 genipv 7507 ltexpri 7611 recexpr 7636 cauappcvgprlemladdru 7654 cauappcvgprlemladdrl 7655 cauappcvgpr 7660 caucvgprlemcl 7674 caucvgprlemladdrl 7676 caucvgpr 7680 caucvgprprlemval 7686 caucvgprprlemnbj 7691 caucvgprprlemmu 7693 caucvgprprlemclphr 7703 caucvgprprlemaddq 7706 caucvgprprlem1 7707 caucvgprprlem2 7708 caucvgsr 7800 pitonnlem1 7843 axi2m1 7873 axcaucvg 7898 |
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