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Theorem opeq1i 4209
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 4206 . 2  |-  ( A  =  B  ->  <. A ,  C >.  =  <. B ,  C >. )
31, 2ax-mp 5 1  |-  <. A ,  C >.  =  <. B ,  C >.
Colors of variables: wff setvar class
Syntax hints:    = wceq 1374   <.cop 4026
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1596  ax-4 1607  ax-5 1675  ax-6 1714  ax-7 1734  ax-10 1781  ax-11 1786  ax-12 1798  ax-13 1961  ax-ext 2438
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 970  df-tru 1377  df-ex 1592  df-nf 1595  df-sb 1707  df-clab 2446  df-cleq 2452  df-clel 2455  df-nfc 2610  df-rab 2816  df-v 3108  df-dif 3472  df-un 3474  df-in 3476  df-ss 3483  df-nul 3779  df-if 3933  df-sn 4021  df-pr 4023  df-op 4027
This theorem is referenced by:  axi2m1  9525  strlemor1  14571  grpbasex  14587  grpplusgx  14588  mat1dimelbas  18733  mat1dim0  18735  mat1dimid  18736  mat1dimscm  18737  mat1dimmul  18738  indistpsx  19270  mapfzcons  30239
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