Theorem eleq1i 2295

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 2292	. 2 |- ( A = B -> ( A e. C <-> B e. C ) )
3	1, 2	ax-mp 5	1 |- ( A e. C <-> B e. C )

Syntax hints:   <-> wb 105   = wceq 1395   e. wcel 2200

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-5 1493  ax-gen 1495  ax-ie1 1539  ax-ie2 1540  ax-4 1556  ax-17 1572  ax-ial 1580  ax-ext 2211

This theorem depends on definitions:  df-bi 117  df-cleq 2222  df-clel 2225

This theorem is referenced by:  eleq12i  2297  eqeltri  2302  intexrabim  4237  abssexg  4266  abnex  4538  snnex  4539  pwexb  4565  sucexb  4589  omex  4685  iprc  4993  dfse2  5101  fressnfv  5826  fnotovb  6047  f1stres  6305  f2ndres  6306  ottposg  6401  dftpos4  6409  frecabex  6544  oacl  6606  diffifi  7056  djuexb  7211  pitonn  8035  axicn  8050  pnfnre  8188  mnfnre  8189  0mnnnnn0  9401  pfxccatin12lem3  11264  pfxccat3  11266  swrdccat  11267  pfxccat3a  11270  swrdccat3blem  11271  swrdccat3b  11272  nprmi  12646  issubm  13505  issrg  13928  srgfcl  13936  subrngrng  14166  txdis1cn  14952  xmeterval  15109  expcncf  15283  gausslemma2dlem1a  15737  2lgslem4  15782  bj-sucexg  16285