Theorem eleq1i 2144

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 2141	. 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 1284   e. wcel 1433

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 1376  ax-gen 1378  ax-ie1 1422  ax-ie2 1423  ax-4 1440  ax-17 1459  ax-ial 1467  ax-ext 2063

This theorem depends on definitions:  df-bi 115  df-cleq 2074  df-clel 2077

This theorem is referenced by:  eleq12i  2146  eqeltri  2151  intexrabim  3928  abssexg  3955  snnex  4199  pwexb  4224  sucexb  4241  omex  4334  iprc  4618  dfse2  4718  fressnfv  5371  fnotovb  5568  f1stres  5806  f2ndres  5807  ottposg  5893  dftpos4  5901  frecabex  6007  oacl  6063  diffifi  6378  pitonn  7016  axicn  7031  pnfnre  7160  mnfnre  7161  0mnnnnn0  8320  nprmi  10506  bj-sucexg  10713