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Theorem opeq12i 3931
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 3928 . 2  |-  ( ( A  =  B  /\  C  =  D )  -> 
<. A ,  C >.  = 
<. B ,  D >. )
41, 2, 3mp2an 654 1  |-  <. A ,  C >.  =  <. B ,  D >.
Colors of variables: wff set class
Syntax hints:    = wceq 1649   <.cop 3760
This theorem is referenced by:  elxp6  6317  addcompq  8760  mulcompq  8762  addassnq  8768  mulassnq  8769  distrnq  8771  1lt2nq  8783  axi2m1  8967  om2uzrdg  11223  rngoi  21816  nvop2  21935  nvvop  21936  phop  22167  hhsssh  22617  axlowdimlem6  25600  isdrngo1  26263
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1552  ax-5 1563  ax-17 1623  ax-9 1661  ax-8 1682  ax-6 1736  ax-7 1741  ax-11 1753  ax-12 1939  ax-ext 2368
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3an 938  df-tru 1325  df-ex 1548  df-nf 1551  df-sb 1656  df-clab 2374  df-cleq 2380  df-clel 2383  df-nfc 2512  df-rab 2658  df-v 2901  df-dif 3266  df-un 3268  df-in 3270  df-ss 3277  df-nul 3572  df-if 3683  df-sn 3763  df-pr 3764  df-op 3766
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