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  4272  epse  4433  unex  4532  ordtri2orexmid  4615  onsucsssucexmid  4619  ordsoexmid  4654  ordtri2or2exmid  4663  ontri2orexmidim  4664  nnregexmid  4713  abrexex  6268  opabex3  6273  abrexex2  6275  abexssex  6276  abexex  6277  oprabrexex2  6281  tfr0dm  6474  exmidonfinlem  7382  1lt2pi  7538  prarloclemarch2  7617  prarloclemlt  7691  0cn  8149  resubcli  8420  0reALT  8454  10nn  9604  numsucc  9628  nummac  9633  qreccl  9849  unirnioo  10181  fz0to4untppr  10332  cats1fvn  11312  4sqlem19  12948  dec2dvds  12950  modsubi  12958  gcdi  12959  prdsex  13318  fn0g  13424  fngsum  13437  sn0topon  14778  retopbas  15213  blssioo  15243  hovercncf  15336  lgslem4  15698  bj-unex  16365  exmidsbthrlem  16478