Theorem eqeltrri 2267

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 2197	. 2 |- B = A
3		eqeltrr.2	. 2 |- A e. C
4	2, 3	eqeltri 2266	1 |- B e. C

Syntax hints:   = wceq 1364   e. wcel 2164

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 1458  ax-gen 1460  ax-ie1 1504  ax-ie2 1505  ax-4 1521  ax-17 1537  ax-ial 1545  ax-ext 2175

This theorem depends on definitions:  df-bi 117  df-cleq 2186  df-clel 2189

This theorem is referenced by:  3eltr3i  2274  p0ex  4218  epse  4374  unex  4473  ordtri2orexmid  4556  onsucsssucexmid  4560  ordsoexmid  4595  ordtri2or2exmid  4604  ontri2orexmidim  4605  nnregexmid  4654  abrexex  6171  opabex3  6176  abrexex2  6178  abexssex  6179  abexex  6180  oprabrexex2  6184  tfr0dm  6377  exmidonfinlem  7255  1lt2pi  7402  prarloclemarch2  7481  prarloclemlt  7555  0cn  8013  resubcli  8284  0reALT  8318  10nn  9466  numsucc  9490  nummac  9495  qreccl  9710  unirnioo  10042  fz0to4untppr  10193  4sqlem19  12550  prdsex  12883  fn0g  12961  fngsum  12974  sn0topon  14267  retopbas  14702  blssioo  14732  hovercncf  14825  lgslem4  15160  bj-unex  15481  exmidsbthrlem  15582