Theorem eqeltrri 2279

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 2209	. 2 |- B = A
3		eqeltrr.2	. 2 |- A e. C
4	2, 3	eqeltri 2278	1 |- B e. C

Syntax hints:   = wceq 1373   e. wcel 2176

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 1470  ax-gen 1472  ax-ie1 1516  ax-ie2 1517  ax-4 1533  ax-17 1549  ax-ial 1557  ax-ext 2187

This theorem depends on definitions:  df-bi 117  df-cleq 2198  df-clel 2201

This theorem is referenced by:  3eltr3i  2286  p0ex  4233  epse  4390  unex  4489  ordtri2orexmid  4572  onsucsssucexmid  4576  ordsoexmid  4611  ordtri2or2exmid  4620  ontri2orexmidim  4621  nnregexmid  4670  abrexex  6204  opabex3  6209  abrexex2  6211  abexssex  6212  abexex  6213  oprabrexex2  6217  tfr0dm  6410  exmidonfinlem  7303  1lt2pi  7455  prarloclemarch2  7534  prarloclemlt  7608  0cn  8066  resubcli  8337  0reALT  8371  10nn  9521  numsucc  9545  nummac  9550  qreccl  9765  unirnioo  10097  fz0to4untppr  10248  4sqlem19  12765  dec2dvds  12767  modsubi  12775  gcdi  12776  prdsex  13134  fn0g  13240  fngsum  13253  sn0topon  14593  retopbas  15028  blssioo  15058  hovercncf  15151  lgslem4  15513  bj-unex  15892  exmidsbthrlem  15998