Theorem eqeltrri 2312

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 2242	. 2 |- B = A
3		eqeltrr.2	. 2 |- A e. C
4	2, 3	eqeltri 2311	1 |- B e. C

Syntax hints:   = wceq 1402   e. wcel 2209

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 1500  ax-gen 1502  ax-ie1 1546  ax-ie2 1547  ax-4 1563  ax-17 1579  ax-ial 1587  ax-ext 2220

This theorem depends on definitions:  df-bi 117  df-cleq 2231  df-clel 2234

This theorem is referenced by:  3eltr3i  2319  p0ex  4320  epse  4482  unex  4582  ordtri2orexmid  4665  onsucsssucexmid  4669  ordsoexmid  4704  ordtri2or2exmid  4713  ontri2orexmidim  4714  nnregexmid  4763  abrexex  6336  opabex3  6341  abrexex2  6343  abexssex  6344  abexex  6345  oprabrexex2  6353  tfr0dm  6583  exmidonfinlem  7535  1lt2pi  7697  prarloclemarch2  7776  prarloclemlt  7850  0cn  8308  resubcli  8579  0reALT  8613  10nn  9771  numsucc  9795  nummac  9800  qreccl  10021  unirnioo  10354  fz0to4untppr  10509  cats1fvn  11514  4sqlem19  13166  dec2dvds  13168  modsubi  13176  gcdi  13177  ballotfilemth  13259  fn0g  13672  fngzsum  13685  prdsex  14149  sn0topon  15112  retopbas  15547  blssioo  15577  hovercncf  15670  lgslem4  16036  konigsberglem1  16643  bj-unex  16859  exmidsbthrlem  16972