Theorem eqeltrri 2308

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 2238	. 2 |- B = A
3		eqeltrr.2	. 2 |- A e. C
4	2, 3	eqeltri 2307	1 |- B e. C

Syntax hints:   = wceq 1398   e. wcel 2205

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 1496  ax-gen 1498  ax-ie1 1542  ax-ie2 1543  ax-4 1559  ax-17 1575  ax-ial 1583  ax-ext 2216

This theorem depends on definitions:  df-bi 117  df-cleq 2227  df-clel 2230

This theorem is referenced by:  3eltr3i  2315  p0ex  4303  epse  4465  unex  4564  ordtri2orexmid  4647  onsucsssucexmid  4651  ordsoexmid  4686  ordtri2or2exmid  4695  ontri2orexmidim  4696  nnregexmid  4745  abrexex  6312  opabex3  6317  abrexex2  6319  abexssex  6320  abexex  6321  oprabrexex2  6325  tfr0dm  6555  exmidonfinlem  7498  1lt2pi  7657  prarloclemarch2  7736  prarloclemlt  7810  0cn  8268  resubcli  8538  0reALT  8572  10nn  9727  numsucc  9751  nummac  9756  qreccl  9977  unirnioo  10309  fz0to4untppr  10462  cats1fvn  11460  4sqlem19  13111  dec2dvds  13113  modsubi  13121  gcdi  13122  prdsex  13499  fn0g  13605  fngsum  13618  sn0topon  14970  retopbas  15405  blssioo  15435  hovercncf  15528  lgslem4  15893  konigsberglem1  16500  bj-unex  16706  exmidsbthrlem  16819