Theorem eleq1i 2100

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 2097	. 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 98   = wceq 1242   e. wcel 1390

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 99  ax-ia2 100  ax-ia3 101  ax-5 1333  ax-gen 1335  ax-ie1 1379  ax-ie2 1380  ax-4 1397  ax-17 1416  ax-ial 1424  ax-ext 2019

This theorem depends on definitions:  df-bi 110  df-cleq 2030  df-clel 2033

This theorem is referenced by:  eleq12i  2102  eqeltri  2107  intexrabim  3898  abssexg  3925  snnex  4147  pwexb  4172  sucexb  4189  omex  4259  iprc  4543  dfse2  4641  fressnfv  5293  fnotovb  5490  f1stres  5728  f2ndres  5729  ottposg  5811  dftpos4  5819  frecabex  5923  oacl  5979  pitonn  6744  axicn  6749  pnfnre  6864  mnfnre  6865  0mnnnnn0  7990  bj-sucexg  9377