Theorem eqeltrri 2251

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 2181	. 2 |- B = A
3		eqeltrr.2	. 2 |- A e. C
4	2, 3	eqeltri 2250	1 |- B e. C

Syntax hints:   = wceq 1353   e. wcel 2148

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 1447  ax-gen 1449  ax-ie1 1493  ax-ie2 1494  ax-4 1510  ax-17 1526  ax-ial 1534  ax-ext 2159

This theorem depends on definitions:  df-bi 117  df-cleq 2170  df-clel 2173

This theorem is referenced by:  3eltr3i  2258  p0ex  4188  epse  4342  unex  4441  ordtri2orexmid  4522  onsucsssucexmid  4526  ordsoexmid  4561  ordtri2or2exmid  4570  ontri2orexmidim  4571  nnregexmid  4620  abrexex  6117  opabex3  6122  abrexex2  6124  abexssex  6125  abexex  6126  oprabrexex2  6130  tfr0dm  6322  exmidonfinlem  7191  1lt2pi  7338  prarloclemarch2  7417  prarloclemlt  7491  0cn  7948  resubcli  8219  0reALT  8253  10nn  9398  numsucc  9422  nummac  9427  qreccl  9641  unirnioo  9972  fz0to4untppr  10123  prdsex  12717  fn0g  12793  sn0topon  13558  retopbas  13993  blssioo  14015  lgslem4  14374  bj-unex  14641  exmidsbthrlem  14740