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Theorem opeq12i 3981
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 3978 . 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 1652   <.cop 3809
This theorem is referenced by:  elxp6  6370  addcompq  8819  mulcompq  8821  addassnq  8827  mulassnq  8828  distrnq  8830  1lt2nq  8842  axi2m1  9026  om2uzrdg  11288  rngoi  21960  nvop2  22079  nvvop  22080  phop  22311  hhsssh  22761  axlowdimlem6  25878  isdrngo1  26563
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1555  ax-5 1566  ax-17 1626  ax-9 1666  ax-8 1687  ax-6 1744  ax-7 1749  ax-11 1761  ax-12 1950  ax-ext 2416
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3an 938  df-tru 1328  df-ex 1551  df-nf 1554  df-sb 1659  df-clab 2422  df-cleq 2428  df-clel 2431  df-nfc 2560  df-rab 2706  df-v 2950  df-dif 3315  df-un 3317  df-in 3319  df-ss 3326  df-nul 3621  df-if 3732  df-sn 3812  df-pr 3813  df-op 3815
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