Theorem eleq1i 2262

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 2259	. 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 1364   e. wcel 2167

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 1461  ax-gen 1463  ax-ie1 1507  ax-ie2 1508  ax-4 1524  ax-17 1540  ax-ial 1548  ax-ext 2178

This theorem depends on definitions:  df-bi 117  df-cleq 2189  df-clel 2192

This theorem is referenced by:  eleq12i  2264  eqeltri  2269  intexrabim  4186  abssexg  4215  abnex  4482  snnex  4483  pwexb  4509  sucexb  4533  omex  4629  iprc  4934  dfse2  5042  fressnfv  5749  fnotovb  5965  f1stres  6217  f2ndres  6218  ottposg  6313  dftpos4  6321  frecabex  6456  oacl  6518  diffifi  6955  djuexb  7110  pitonn  7915  axicn  7930  pnfnre  8068  mnfnre  8069  0mnnnnn0  9281  nprmi  12292  issubm  13104  issrg  13521  srgfcl  13529  subrngrng  13758  txdis1cn  14514  xmeterval  14671  expcncf  14845  gausslemma2dlem1a  15299  2lgslem4  15344  bj-sucexg  15568