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Theorem eqeq12i 2095
Description: A useful inference for substituting definitions into an equality. (Contributed by NM, 5-Aug-1993.) (Proof shortened by Andrew Salmon, 25-May-2011.)
Hypotheses
Ref Expression
eqeq12i.1  |-  A  =  B
eqeq12i.2  |-  C  =  D
Assertion
Ref Expression
eqeq12i  |-  ( A  =  C  <->  B  =  D )

Proof of Theorem eqeq12i
StepHypRef Expression
1 eqeq12i.1 . 2  |-  A  =  B
2 eqeq12i.2 . 2  |-  C  =  D
3 eqeq12 2094 . 2  |-  ( ( A  =  B  /\  C  =  D )  ->  ( A  =  C  <-> 
B  =  D ) )
41, 2, 3mp2an 417 1  |-  ( A  =  C  <->  B  =  D )
Colors of variables: wff set class
Syntax hints:    <-> wb 103    = wceq 1285
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106  ax-5 1377  ax-gen 1379  ax-4 1441  ax-17 1460  ax-ext 2064
This theorem depends on definitions:  df-bi 115  df-cleq 2075
This theorem is referenced by:  rabbi  2532  sbceqg  2923  preqr2g  3567  preqr2  3569  otth  4005  rncoeq  4633  eqfnov  5638  mpt22eqb  5641  f1o2ndf1  5880  ecopovsym  6268  sq11i  9662
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