Theorem eleq1i 2145

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

Step	Hyp	Ref	Expression
1		eleq1i.1	. 2 |- A = B
2		eleq1 2142	. 2 |- ( A = B -> ( A e. C <-> B e. C ) )
3	1, 2	ax-mp 7	1 |- ( A e. C <-> B e. C )

Syntax hints:   <-> wb 103   = wceq 1285   e. wcel 1434

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-ie1 1423  ax-ie2 1424  ax-4 1441  ax-17 1460  ax-ial 1468  ax-ext 2064

This theorem depends on definitions:  df-bi 115  df-cleq 2075  df-clel 2078

This theorem is referenced by:  eleq12i  2147  eqeltri  2152  intexrabim  3936  abssexg  3963  snnex  4207  pwexb  4232  sucexb  4249  omex  4342  iprc  4628  dfse2  4728  fressnfv  5382  fnotovb  5579  f1stres  5817  f2ndres  5818  ottposg  5904  dftpos4  5912  frecabex  6047  oacl  6104  diffifi  6428  pitonn  7078  axicn  7093  pnfnre  7222  mnfnre  7223  0mnnnnn0  8387  nprmi  10650  bj-sucexg  10898