Theorem eqeltrri 2305

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 2235	. 2 |- B = A
3		eqeltrr.2	. 2 |- A e. C
4	2, 3	eqeltri 2304	1 |- B e. C

Syntax hints:   = wceq 1397   e. wcel 2202

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 1495  ax-gen 1497  ax-ie1 1541  ax-ie2 1542  ax-4 1558  ax-17 1574  ax-ial 1582  ax-ext 2213

This theorem depends on definitions:  df-bi 117  df-cleq 2224  df-clel 2227

This theorem is referenced by:  3eltr3i  2312  p0ex  4278  epse  4439  unex  4538  ordtri2orexmid  4621  onsucsssucexmid  4625  ordsoexmid  4660  ordtri2or2exmid  4669  ontri2orexmidim  4670  nnregexmid  4719  abrexex  6278  opabex3  6283  abrexex2  6285  abexssex  6286  abexex  6287  oprabrexex2  6291  tfr0dm  6487  exmidonfinlem  7403  1lt2pi  7559  prarloclemarch2  7638  prarloclemlt  7712  0cn  8170  resubcli  8441  0reALT  8475  10nn  9625  numsucc  9649  nummac  9654  qreccl  9875  unirnioo  10207  fz0to4untppr  10358  cats1fvn  11344  4sqlem19  12981  dec2dvds  12983  modsubi  12991  gcdi  12992  prdsex  13351  fn0g  13457  fngsum  13470  sn0topon  14811  retopbas  15246  blssioo  15276  hovercncf  15369  lgslem4  15731  bj-unex  16514  exmidsbthrlem  16626