ILE Home Intuitionistic Logic Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  ILE Home  >  Th. List  >  opeq2i Unicode version

Theorem opeq2i 3709
Description: Equality inference for ordered pairs. (Contributed by NM, 16-Dec-2006.)
Hypothesis
Ref Expression
opeq1i.1  |-  A  =  B
Assertion
Ref Expression
opeq2i  |-  <. C ,  A >.  =  <. C ,  B >.

Proof of Theorem opeq2i
StepHypRef Expression
1 opeq1i.1 . 2  |-  A  =  B
2 opeq2 3706 . 2  |-  ( A  =  B  ->  <. C ,  A >.  =  <. C ,  B >. )
31, 2ax-mp 5 1  |-  <. C ,  A >.  =  <. C ,  B >.
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:  fnressn  5606  fressnfv  5607  nqprlu  7367  suplocexpr  7545  addresr  7657  iseqvalcbv  10242
  Copyright terms: Public domain W3C validator