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Theorem opeq12i 3710
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 3707 . 2  |-  ( ( A  =  B  /\  C  =  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 3530
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 2121
This theorem depends on definitions:  df-bi 116  df-3an 964  df-tru 1334  df-nf 1437  df-sb 1736  df-clab 2126  df-cleq 2132  df-clel 2135  df-nfc 2270  df-v 2688  df-un 3075  df-sn 3533  df-pr 3534  df-op 3536
This theorem is referenced by:  addpinq1  7284  genipv  7329  ltexpri  7433  recexpr  7458  cauappcvgprlemladdru  7476  cauappcvgprlemladdrl  7477  cauappcvgpr  7482  caucvgprlemcl  7496  caucvgprlemladdrl  7498  caucvgpr  7502  caucvgprprlemval  7508  caucvgprprlemnbj  7513  caucvgprprlemmu  7515  caucvgprprlemclphr  7525  caucvgprprlemaddq  7528  caucvgprprlem1  7529  caucvgprprlem2  7530  caucvgsr  7622  pitonnlem1  7665  axi2m1  7695  axcaucvg  7720
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