Theorem eqeltrri 2263

Description: Substitution of equal classes into membership relation. (Contributed by NM, 5-Aug-1993.)

Hypotheses

Ref	Expression
eqeltrr.1	⊢ 𝐴 = 𝐵
eqeltrr.2	⊢ 𝐴 ∈ 𝐶

Assertion

Ref	Expression
eqeltrri	⊢ 𝐵 ∈ 𝐶

Proof of Theorem eqeltrri

Step	Hyp	Ref	Expression
1		eqeltrr.1	. . 3 ⊢ 𝐴 = 𝐵
2	1	eqcomi 2193	. 2 ⊢ 𝐵 = 𝐴
3		eqeltrr.2	. 2 ⊢ 𝐴 ∈ 𝐶
4	2, 3	eqeltri 2262	1 ⊢ 𝐵 ∈ 𝐶

Syntax hints:   = wceq 1364   ∈ wcel 2160

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 1458  ax-gen 1460  ax-ie1 1504  ax-ie2 1505  ax-4 1521  ax-17 1537  ax-ial 1545  ax-ext 2171

This theorem depends on definitions:  df-bi 117  df-cleq 2182  df-clel 2185

This theorem is referenced by:  3eltr3i  2270  p0ex  4206  epse  4360  unex  4459  ordtri2orexmid  4540  onsucsssucexmid  4544  ordsoexmid  4579  ordtri2or2exmid  4588  ontri2orexmidim  4589  nnregexmid  4638  abrexex  6142  opabex3  6147  abrexex2  6149  abexssex  6150  abexex  6151  oprabrexex2  6155  tfr0dm  6347  exmidonfinlem  7222  1lt2pi  7369  prarloclemarch2  7448  prarloclemlt  7522  0cn  7979  resubcli  8250  0reALT  8284  10nn  9429  numsucc  9453  nummac  9458  qreccl  9672  unirnioo  10003  fz0to4untppr  10154  4sqlem19  12441  prdsex  12774  fn0g  12851  sn0topon  14048  retopbas  14483  blssioo  14505  lgslem4  14865  bj-unex  15132  exmidsbthrlem  15232