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Theorem opeq12i 3622
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 3619 . 2 ((𝐴 = 𝐵𝐶 = 𝐷) → ⟨𝐴, 𝐶⟩ = ⟨𝐵, 𝐷⟩)
41, 2, 3mp2an 417 1 𝐴, 𝐶⟩ = ⟨𝐵, 𝐷
Colors of variables: wff set class
Syntax hints:   = wceq 1289  cop 3444
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106  ax-io 665  ax-5 1381  ax-7 1382  ax-gen 1383  ax-ie1 1427  ax-ie2 1428  ax-8 1440  ax-10 1441  ax-11 1442  ax-i12 1443  ax-bndl 1444  ax-4 1445  ax-17 1464  ax-i9 1468  ax-ial 1472  ax-i5r 1473  ax-ext 2070
This theorem depends on definitions:  df-bi 115  df-3an 926  df-tru 1292  df-nf 1395  df-sb 1693  df-clab 2075  df-cleq 2081  df-clel 2084  df-nfc 2217  df-v 2621  df-un 3001  df-sn 3447  df-pr 3448  df-op 3450
This theorem is referenced by:  addpinq1  7002  genipv  7047  ltexpri  7151  recexpr  7176  cauappcvgprlemladdru  7194  cauappcvgprlemladdrl  7195  cauappcvgpr  7200  caucvgprlemcl  7214  caucvgprlemladdrl  7216  caucvgpr  7220  caucvgprprlemval  7226  caucvgprprlemnbj  7231  caucvgprprlemmu  7233  caucvgprprlemclphr  7243  caucvgprprlemaddq  7246  caucvgprprlem1  7247  caucvgprprlem2  7248  caucvgsr  7326  pitonnlem1  7361  axi2m1  7389  axcaucvg  7414
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