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Theorem opeq1i 3974
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 3971 . 2  |-  ( A  =  B  ->  <. A ,  C >.  =  <. B ,  C >. )
31, 2ax-mp 8 1  |-  <. A ,  C >.  =  <. B ,  C >.
Colors of variables: wff set class
Syntax hints:    = wceq 1652   <.cop 3804
This theorem is referenced by:  axi2m1  9018  strlemor1  13539  grpbasex  13555  grpplusgx  13556  indistpsx  17057  mapfzcons  26704
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 2411
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 2417  df-cleq 2423  df-clel 2426  df-nfc 2555  df-rab 2701  df-v 2945  df-dif 3310  df-un 3312  df-in 3314  df-ss 3321  df-nul 3616  df-if 3727  df-sn 3807  df-pr 3808  df-op 3810
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