Theorem eqeltrri 2303

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 2233	. 2 ⊢ 𝐵 = 𝐴
3		eqeltrr.2	. 2 ⊢ 𝐴 ∈ 𝐶
4	2, 3	eqeltri 2302	1 ⊢ 𝐵 ∈ 𝐶

Syntax hints:   = wceq 1395   ∈ wcel 2200

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 1493  ax-gen 1495  ax-ie1 1539  ax-ie2 1540  ax-4 1556  ax-17 1572  ax-ial 1580  ax-ext 2211

This theorem depends on definitions:  df-bi 117  df-cleq 2222  df-clel 2225

This theorem is referenced by:  3eltr3i  2310  p0ex  4272  epse  4433  unex  4532  ordtri2orexmid  4615  onsucsssucexmid  4619  ordsoexmid  4654  ordtri2or2exmid  4663  ontri2orexmidim  4664  nnregexmid  4713  abrexex  6262  opabex3  6267  abrexex2  6269  abexssex  6270  abexex  6271  oprabrexex2  6275  tfr0dm  6468  exmidonfinlem  7371  1lt2pi  7527  prarloclemarch2  7606  prarloclemlt  7680  0cn  8138  resubcli  8409  0reALT  8443  10nn  9593  numsucc  9617  nummac  9622  qreccl  9837  unirnioo  10169  fz0to4untppr  10320  cats1fvn  11296  4sqlem19  12932  dec2dvds  12934  modsubi  12942  gcdi  12943  prdsex  13302  fn0g  13408  fngsum  13421  sn0topon  14762  retopbas  15197  blssioo  15227  hovercncf  15320  lgslem4  15682  bj-unex  16282  exmidsbthrlem  16390