Theorem eqeltrri 2267

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 2197	. 2 ⊢ 𝐵 = 𝐴
3		eqeltrr.2	. 2 ⊢ 𝐴 ∈ 𝐶
4	2, 3	eqeltri 2266	1 ⊢ 𝐵 ∈ 𝐶

Syntax hints:   = wceq 1364   ∈ wcel 2164

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 2175

This theorem depends on definitions:  df-bi 117  df-cleq 2186  df-clel 2189

This theorem is referenced by:  3eltr3i  2274  p0ex  4217  epse  4373  unex  4472  ordtri2orexmid  4555  onsucsssucexmid  4559  ordsoexmid  4594  ordtri2or2exmid  4603  ontri2orexmidim  4604  nnregexmid  4653  abrexex  6169  opabex3  6174  abrexex2  6176  abexssex  6177  abexex  6178  oprabrexex2  6182  tfr0dm  6375  exmidonfinlem  7253  1lt2pi  7400  prarloclemarch2  7479  prarloclemlt  7553  0cn  8011  resubcli  8282  0reALT  8316  10nn  9463  numsucc  9487  nummac  9492  qreccl  9707  unirnioo  10039  fz0to4untppr  10190  4sqlem19  12547  prdsex  12880  fn0g  12958  fngsum  12971  sn0topon  14256  retopbas  14691  blssioo  14713  hovercncf  14800  lgslem4  15119  bj-unex  15411  exmidsbthrlem  15512