Theorem eqeltrri 2305

Description: Substitution of equal classes into membership relation. (Contributed by NM, 5-Aug-1993.)

Hypotheses

Ref	Expression
eqeltrr.1	|- A = B
eqeltrr.2	|- A e. C

Assertion

Ref	Expression
eqeltrri	|- B e. C

Proof of Theorem eqeltrri

Step	Hyp	Ref	Expression
1		eqeltrr.1	. . 3 |- A = B
2	1	eqcomi 2235	. 2 |- B = A
3		eqeltrr.2	. 2 |- A e. C
4	2, 3	eqeltri 2304	1 |- B e. C

Syntax hints:   = wceq 1398   e. wcel 2202

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 2213

This theorem depends on definitions:  df-bi 117  df-cleq 2224  df-clel 2227

This theorem is referenced by:  3eltr3i  2312  p0ex  4284  epse  4445  unex  4544  ordtri2orexmid  4627  onsucsssucexmid  4631  ordsoexmid  4666  ordtri2or2exmid  4675  ontri2orexmidim  4676  nnregexmid  4725  abrexex  6288  opabex3  6293  abrexex2  6295  abexssex  6296  abexex  6297  oprabrexex2  6301  tfr0dm  6531  exmidonfinlem  7464  1lt2pi  7620  prarloclemarch2  7699  prarloclemlt  7773  0cn  8231  resubcli  8501  0reALT  8535  10nn  9687  numsucc  9711  nummac  9716  qreccl  9937  unirnioo  10269  fz0to4untppr  10421  cats1fvn  11411  4sqlem19  13062  dec2dvds  13064  modsubi  13072  gcdi  13073  prdsex  13432  fn0g  13538  fngsum  13551  sn0topon  14899  retopbas  15334  blssioo  15364  hovercncf  15457  lgslem4  15822  konigsberglem1  16429  bj-unex  16635  exmidsbthrlem  16750