Theorem eleq1i 2255

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 2252	. 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 2160

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 1458  ax-gen 1460  ax-ie1 1504  ax-ie2 1505  ax-4 1521  ax-17 1537  ax-ial 1545  ax-ext 2171

This theorem depends on definitions:  df-bi 117  df-cleq 2182  df-clel 2185

This theorem is referenced by:  eleq12i  2257  eqeltri  2262  intexrabim  4168  abssexg  4197  abnex  4462  snnex  4463  pwexb  4489  sucexb  4511  omex  4607  iprc  4910  dfse2  5016  fressnfv  5719  fnotovb  5934  f1stres  6178  f2ndres  6179  ottposg  6274  dftpos4  6282  frecabex  6417  oacl  6479  diffifi  6912  djuexb  7061  pitonn  7865  axicn  7880  pnfnre  8017  mnfnre  8018  0mnnnnn0  9226  nprmi  12142  issubm  12890  issrg  13280  srgfcl  13288  subrngrng  13510  txdis1cn  14175  xmeterval  14332  expcncf  14489  bj-sucexg  15071