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  4273  epse  4434  unex  4533  ordtri2orexmid  4616  onsucsssucexmid  4620  ordsoexmid  4655  ordtri2or2exmid  4664  ontri2orexmidim  4665  nnregexmid  4714  abrexex  6271  opabex3  6276  abrexex2  6278  abexssex  6279  abexex  6280  oprabrexex2  6284  tfr0dm  6479  exmidonfinlem  7387  1lt2pi  7543  prarloclemarch2  7622  prarloclemlt  7696  0cn  8154  resubcli  8425  0reALT  8459  10nn  9609  numsucc  9633  nummac  9638  qreccl  9854  unirnioo  10186  fz0to4untppr  10337  cats1fvn  11317  4sqlem19  12953  dec2dvds  12955  modsubi  12963  gcdi  12964  prdsex  13323  fn0g  13429  fngsum  13442  sn0topon  14783  retopbas  15218  blssioo  15248  hovercncf  15341  lgslem4  15703  bj-unex  16391  exmidsbthrlem  16504