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Theorem opeq12i 3705
 Description: Equality inference for ordered pairs. (Contributed by NM, 16-Dec-2006.) (Proof shortened by Eric Schmidt, 4-Apr-2007.)
Hypotheses
Ref Expression
opeq1i.1 𝐴 = 𝐵
opeq12i.2 𝐶 = 𝐷
Assertion
Ref Expression
opeq12i 𝐴, 𝐶⟩ = ⟨𝐵, 𝐷

Proof of Theorem opeq12i
StepHypRef Expression
1 opeq1i.1 . 2 𝐴 = 𝐵
2 opeq12i.2 . 2 𝐶 = 𝐷
3 opeq12 3702 . 2 ((𝐴 = 𝐵𝐶 = 𝐷) → ⟨𝐴, 𝐶⟩ = ⟨𝐵, 𝐷⟩)
41, 2, 3mp2an 422 1 𝐴, 𝐶⟩ = ⟨𝐵, 𝐷
 Colors of variables: wff set class Syntax hints:   = wceq 1331  ⟨cop 3525 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 2119 This theorem depends on definitions:  df-bi 116  df-3an 964  df-tru 1334  df-nf 1437  df-sb 1736  df-clab 2124  df-cleq 2130  df-clel 2133  df-nfc 2268  df-v 2683  df-un 3070  df-sn 3528  df-pr 3529  df-op 3531 This theorem is referenced by:  addpinq1  7265  genipv  7310  ltexpri  7414  recexpr  7439  cauappcvgprlemladdru  7457  cauappcvgprlemladdrl  7458  cauappcvgpr  7463  caucvgprlemcl  7477  caucvgprlemladdrl  7479  caucvgpr  7483  caucvgprprlemval  7489  caucvgprprlemnbj  7494  caucvgprprlemmu  7496  caucvgprprlemclphr  7506  caucvgprprlemaddq  7509  caucvgprprlem1  7510  caucvgprprlem2  7511  caucvgsr  7603  pitonnlem1  7646  axi2m1  7676  axcaucvg  7701
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