Theorem eqeltrri 2152

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 2085	. 2 |- B = A
3		eqeltrr.2	. 2 |- A e. C
4	2, 3	eqeltri 2151	1 |- B e. C

Syntax hints:   = wceq 1284   e. wcel 1433

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106  ax-5 1376  ax-gen 1378  ax-ie1 1422  ax-ie2 1423  ax-4 1440  ax-17 1459  ax-ial 1467  ax-ext 2063

This theorem depends on definitions:  df-bi 115  df-cleq 2074  df-clel 2077

This theorem is referenced by:  3eltr3i  2159  p0ex  3959  epse  4097  unex  4194  ordtri2orexmid  4266  onsucsssucexmid  4270  ordsoexmid  4305  ordtri2or2exmid  4314  nnregexmid  4360  abrexex  5764  opabex3  5769  abrexex2  5771  abexssex  5772  abexex  5773  oprabrexex2  5777  tfr0  5960  1lt2pi  6530  prarloclemarch2  6609  prarloclemlt  6683  0cn  7111  resubcli  7371  0reALT  7405  10nn  8492  numsucc  8516  nummac  8521  qreccl  8727  unirnioo  8996  bj-unex  10710