Theorem eqeltrri 2303

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 2233	. 2 |- B = A
3		eqeltrr.2	. 2 |- A e. C
4	2, 3	eqeltri 2302	1 |- B e. C

Syntax hints:   = wceq 1395   e. wcel 2200

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 1493  ax-gen 1495  ax-ie1 1539  ax-ie2 1540  ax-4 1556  ax-17 1572  ax-ial 1580  ax-ext 2211

This theorem depends on definitions:  df-bi 117  df-cleq 2222  df-clel 2225

This theorem is referenced by:  3eltr3i  2310  p0ex  4276  epse  4437  unex  4536  ordtri2orexmid  4619  onsucsssucexmid  4623  ordsoexmid  4658  ordtri2or2exmid  4667  ontri2orexmidim  4668  nnregexmid  4717  abrexex  6274  opabex3  6279  abrexex2  6281  abexssex  6282  abexex  6283  oprabrexex2  6287  tfr0dm  6483  exmidonfinlem  7394  1lt2pi  7550  prarloclemarch2  7629  prarloclemlt  7703  0cn  8161  resubcli  8432  0reALT  8466  10nn  9616  numsucc  9640  nummac  9645  qreccl  9866  unirnioo  10198  fz0to4untppr  10349  cats1fvn  11335  4sqlem19  12972  dec2dvds  12974  modsubi  12982  gcdi  12983  prdsex  13342  fn0g  13448  fngsum  13461  sn0topon  14802  retopbas  15237  blssioo  15267  hovercncf  15360  lgslem4  15722  bj-unex  16450  exmidsbthrlem  16562