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Theorem opeq12i 3582
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 3579 . 2  |-  ( ( A  =  B  /\  C  =  D )  -> 
<. A ,  C >.  = 
<. B ,  D >. )
41, 2, 3mp2an 410 1  |-  <. A ,  C >.  =  <. B ,  D >.
Colors of variables: wff set class
Syntax hints:    = wceq 1259   <.cop 3406
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 103  ax-ia2 104  ax-ia3 105  ax-io 640  ax-5 1352  ax-7 1353  ax-gen 1354  ax-ie1 1398  ax-ie2 1399  ax-8 1411  ax-10 1412  ax-11 1413  ax-i12 1414  ax-bndl 1415  ax-4 1416  ax-17 1435  ax-i9 1439  ax-ial 1443  ax-i5r 1444  ax-ext 2038
This theorem depends on definitions:  df-bi 114  df-3an 898  df-tru 1262  df-nf 1366  df-sb 1662  df-clab 2043  df-cleq 2049  df-clel 2052  df-nfc 2183  df-v 2576  df-un 2950  df-sn 3409  df-pr 3410  df-op 3412
This theorem is referenced by:  addpinq1  6620  genipv  6665  ltexpri  6769  recexpr  6794  cauappcvgprlemladdru  6812  cauappcvgprlemladdrl  6813  cauappcvgpr  6818  caucvgprlemcl  6832  caucvgprlemladdrl  6834  caucvgpr  6838  caucvgprprlemval  6844  caucvgprprlemnbj  6849  caucvgprprlemmu  6851  caucvgprprlemclphr  6861  caucvgprprlemaddq  6864  caucvgprprlem1  6865  caucvgprprlem2  6866  caucvgsr  6944  pitonnlem1  6979  axi2m1  7007  axcaucvg  7032
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