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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 3781 | . 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 3596 |
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 2740 df-un 3134 df-sn 3599 df-pr 3600 df-op 3602 |
This theorem is referenced by: addpinq1 7463 genipv 7508 ltexpri 7612 recexpr 7637 cauappcvgprlemladdru 7655 cauappcvgprlemladdrl 7656 cauappcvgpr 7661 caucvgprlemcl 7675 caucvgprlemladdrl 7677 caucvgpr 7681 caucvgprprlemval 7687 caucvgprprlemnbj 7692 caucvgprprlemmu 7694 caucvgprprlemclphr 7704 caucvgprprlemaddq 7707 caucvgprprlem1 7708 caucvgprprlem2 7709 caucvgsr 7801 pitonnlem1 7844 axi2m1 7874 axcaucvg 7899 |
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