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  4203  epse  4357  unex  4456  ordtri2orexmid  4537  onsucsssucexmid  4541  ordsoexmid  4576  ordtri2or2exmid  4585  ontri2orexmidim  4586  nnregexmid  4635  abrexex  6136  opabex3  6141  abrexex2  6143  abexssex  6144  abexex  6145  oprabrexex2  6149  tfr0dm  6341  exmidonfinlem  7210  1lt2pi  7357  prarloclemarch2  7436  prarloclemlt  7510  0cn  7967  resubcli  8238  0reALT  8272  10nn  9417  numsucc  9441  nummac  9446  qreccl  9660  unirnioo  9991  fz0to4untppr  10142  prdsex  12740  fn0g  12817  sn0topon  13985  retopbas  14420  blssioo  14442  lgslem4  14801  bj-unex  15068  exmidsbthrlem  15168