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Theorem opeq1i 3796
Description: Equality inference for ordered pairs. (Contributed by NM, 16-Dec-2006.)
Hypothesis
Ref Expression
opeq1i.1  |-  A  =  B
Assertion
Ref Expression
opeq1i  |-  <. A ,  C >.  =  <. B ,  C >.

Proof of Theorem opeq1i
StepHypRef Expression
1 opeq1i.1 . 2  |-  A  =  B
2 opeq1 3793 . 2  |-  ( A  =  B  ->  <. A ,  C >.  =  <. B ,  C >. )
31, 2ax-mp 5 1  |-  <. A ,  C >.  =  <. B ,  C >.
Colors of variables: wff set class
Syntax hints:    = wceq 1364   <.cop 3610
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-io 710  ax-5 1458  ax-7 1459  ax-gen 1460  ax-ie1 1504  ax-ie2 1505  ax-8 1515  ax-10 1516  ax-11 1517  ax-i12 1518  ax-bndl 1520  ax-4 1521  ax-17 1537  ax-i9 1541  ax-ial 1545  ax-i5r 1546  ax-ext 2171
This theorem depends on definitions:  df-bi 117  df-3an 982  df-tru 1367  df-nf 1472  df-sb 1774  df-clab 2176  df-cleq 2182  df-clel 2185  df-nfc 2321  df-v 2754  df-un 3148  df-sn 3613  df-pr 3614  df-op 3616
This theorem is referenced by:  caucvgsrlemfv  7820  caucvgsr  7831  pitonnlem1  7874  axi2m1  7904  axcaucvg  7929  ennnfonelem1  12458  2strstr1g  12633  2strop1g  12635
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