Theorem eleq1i 2300

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 2297	. 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 1398   e. wcel 2205

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 1496  ax-gen 1498  ax-ie1 1542  ax-ie2 1543  ax-4 1559  ax-17 1575  ax-ial 1583  ax-ext 2216

This theorem depends on definitions:  df-bi 117  df-cleq 2227  df-clel 2230

This theorem is referenced by:  eleq12i  2302  eqeltri  2307  intexrabim  4270  abssexg  4300  abnex  4573  snnex  4574  pwexb  4600  sucexb  4624  omex  4720  iprc  5031  dfse2  5140  fressnfv  5876  fnotovb  6104  f1stres  6366  f2ndres  6367  ottposg  6499  dftpos4  6507  frecabex  6642  oacl  6706  diffifi  7164  djuexb  7348  pitonn  8179  axicn  8194  pnfnre  8331  mnfnre  8332  0mnnnnn0  9545  fcdmnn0fsupp  9566  pfxccatin12lem3  11449  pfxccat3  11451  swrdccat  11452  pfxccat3a  11455  swrdccat3blem  11456  swrdccat3b  11457  nprmi  12846  issubm  13727  issrg  14208  srgfcl  14216  subrngrng  14448  txdis1cn  15269  xmeterval  15426  expcncf  15600  gausslemma2dlem1a  16057  2lgslem4  16102  clwwlknonex2  16560  bj-sucexg  16818