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Theorem eleq1i 2150
Description: Inference from equality to equivalence of membership. (Contributed by NM, 5-Aug-1993.)
Hypothesis
Ref Expression
eleq1i.1  |-  A  =  B
Assertion
Ref Expression
eleq1i  |-  ( A  e.  C  <->  B  e.  C )

Proof of Theorem eleq1i
StepHypRef Expression
1 eleq1i.1 . 2  |-  A  =  B
2 eleq1 2147 . 2  |-  ( A  =  B  ->  ( A  e.  C  <->  B  e.  C ) )
31, 2ax-mp 7 1  |-  ( A  e.  C  <->  B  e.  C )
Colors of variables: wff set class
Syntax hints:    <-> wb 103    = wceq 1287    e. wcel 1436
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 1379  ax-gen 1381  ax-ie1 1425  ax-ie2 1426  ax-4 1443  ax-17 1462  ax-ial 1470  ax-ext 2067
This theorem depends on definitions:  df-bi 115  df-cleq 2078  df-clel 2081
This theorem is referenced by:  eleq12i  2152  eqeltri  2157  intexrabim  3966  abssexg  3993  snnex  4247  pwexb  4272  sucexb  4289  omex  4383  iprc  4671  dfse2  4774  fressnfv  5449  fnotovb  5651  f1stres  5889  f2ndres  5890  ottposg  5976  dftpos4  5984  frecabex  6119  oacl  6177  diffifi  6564  pitonn  7332  axicn  7347  pnfnre  7476  mnfnre  7477  0mnnnnn0  8641  nprmi  11012  bj-sucexg  11282
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