Theorem eqeltrri 2305

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 2235	. 2 ⊢ 𝐵 = 𝐴
3		eqeltrr.2	. 2 ⊢ 𝐴 ∈ 𝐶
4	2, 3	eqeltri 2304	1 ⊢ 𝐵 ∈ 𝐶

Syntax hints:   = wceq 1397   ∈ wcel 2202

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 1495  ax-gen 1497  ax-ie1 1541  ax-ie2 1542  ax-4 1558  ax-17 1574  ax-ial 1582  ax-ext 2213

This theorem depends on definitions:  df-bi 117  df-cleq 2224  df-clel 2227

This theorem is referenced by:  3eltr3i  2312  p0ex  4278  epse  4439  unex  4538  ordtri2orexmid  4621  onsucsssucexmid  4625  ordsoexmid  4660  ordtri2or2exmid  4669  ontri2orexmidim  4670  nnregexmid  4719  abrexex  6279  opabex3  6284  abrexex2  6286  abexssex  6287  abexex  6288  oprabrexex2  6292  tfr0dm  6488  exmidonfinlem  7404  1lt2pi  7560  prarloclemarch2  7639  prarloclemlt  7713  0cn  8171  resubcli  8442  0reALT  8476  10nn  9626  numsucc  9650  nummac  9655  qreccl  9876  unirnioo  10208  fz0to4untppr  10359  cats1fvn  11346  4sqlem19  12984  dec2dvds  12986  modsubi  12994  gcdi  12995  prdsex  13354  fn0g  13460  fngsum  13473  sn0topon  14815  retopbas  15250  blssioo  15280  hovercncf  15373  lgslem4  15735  bj-unex  16531  exmidsbthrlem  16643