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Theorem eqeq12i 2184
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 2183 . 2  |-  ( ( A  =  B  /\  C  =  D )  ->  ( A  =  C  <-> 
B  =  D ) )
41, 2, 3mp2an 424 1  |-  ( A  =  C  <->  B  =  D )
Colors of variables: wff set class
Syntax hints:    <-> wb 104    = wceq 1348
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-5 1440  ax-gen 1442  ax-4 1503  ax-17 1519  ax-ext 2152
This theorem depends on definitions:  df-bi 116  df-cleq 2163
This theorem is referenced by:  rabbi  2647  sbceqg  3065  preqr2g  3754  preqr2  3756  otth  4227  rncoeq  4884  eqfnov  5959  mpo2eqb  5962  f1o2ndf1  6207  ecopovsym  6609  sq11i  10565  pwle2  14031
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