Theorem eqeltrri 2308

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 2238	. 2 |- B = A
3		eqeltrr.2	. 2 |- A e. C
4	2, 3	eqeltri 2307	1 |- B e. C

Syntax hints:   = 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:  3eltr3i  2315  p0ex  4306  epse  4468  unex  4567  ordtri2orexmid  4650  onsucsssucexmid  4654  ordsoexmid  4689  ordtri2or2exmid  4698  ontri2orexmidim  4699  nnregexmid  4748  abrexex  6319  opabex3  6324  abrexex2  6326  abexssex  6327  abexex  6328  oprabrexex2  6336  tfr0dm  6566  exmidonfinlem  7509  1lt2pi  7671  prarloclemarch2  7750  prarloclemlt  7824  0cn  8282  resubcli  8552  0reALT  8586  10nn  9742  numsucc  9766  nummac  9771  qreccl  9992  unirnioo  10325  fz0to4untppr  10480  cats1fvn  11481  4sqlem19  13132  dec2dvds  13134  modsubi  13142  gcdi  13143  ballotfilemth  13225  fn0g  13638  fngsum  13651  prdsex  14114  sn0topon  15079  retopbas  15514  blssioo  15544  hovercncf  15637  lgslem4  16002  konigsberglem1  16609  bj-unex  16815  exmidsbthrlem  16928