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Theorem opeq12i 3554
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 3551 . 2 ((𝐴 = 𝐵𝐶 = 𝐷) → ⟨𝐴, 𝐶⟩ = ⟨𝐵, 𝐷⟩)
41, 2, 3mp2an 402 1 𝐴, 𝐶⟩ = ⟨𝐵, 𝐷
Colors of variables: wff set class
Syntax hints:   = wceq 1243  cop 3378
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 99  ax-ia2 100  ax-ia3 101  ax-io 630  ax-5 1336  ax-7 1337  ax-gen 1338  ax-ie1 1382  ax-ie2 1383  ax-8 1395  ax-10 1396  ax-11 1397  ax-i12 1398  ax-bndl 1399  ax-4 1400  ax-17 1419  ax-i9 1423  ax-ial 1427  ax-i5r 1428  ax-ext 2022
This theorem depends on definitions:  df-bi 110  df-3an 887  df-tru 1246  df-nf 1350  df-sb 1646  df-clab 2027  df-cleq 2033  df-clel 2036  df-nfc 2167  df-v 2559  df-un 2922  df-sn 3381  df-pr 3382  df-op 3384
This theorem is referenced by:  addpinq1  6560  genipv  6605  ltexpri  6709  recexpr  6734  cauappcvgprlemladdru  6752  cauappcvgprlemladdrl  6753  cauappcvgpr  6758  caucvgprlemcl  6772  caucvgprlemladdrl  6774  caucvgpr  6778  caucvgprprlemval  6784  caucvgprprlemnbj  6789  caucvgprprlemmu  6791  caucvgprprlemclphr  6801  caucvgprprlemaddq  6804  caucvgprprlem1  6805  caucvgprprlem2  6806  caucvgsr  6884  pitonnlem1  6919  axi2m1  6947  axcaucvg  6972
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