Theorem eqeltrri 2312

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 2242	. 2 ⊢ 𝐵 = 𝐴
3		eqeltrr.2	. 2 ⊢ 𝐴 ∈ 𝐶
4	2, 3	eqeltri 2311	1 ⊢ 𝐵 ∈ 𝐶

Syntax hints:   = wceq 1402   ∈ wcel 2209

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 1500  ax-gen 1502  ax-ie1 1546  ax-ie2 1547  ax-4 1563  ax-17 1579  ax-ial 1587  ax-ext 2220

This theorem depends on definitions:  df-bi 117  df-cleq 2231  df-clel 2234

This theorem is referenced by:  3eltr3i  2319  p0ex  4323  epse  4485  unex  4585  ordtri2orexmid  4668  onsucsssucexmid  4672  ordsoexmid  4707  ordtri2or2exmid  4716  ontri2orexmidim  4717  nnregexmid  4766  abrexex  6340  opabex3  6345  abrexex2  6347  abexssex  6348  abexex  6349  oprabrexex2  6357  tfr0dm  6587  exmidonfinlem  7539  1lt2pi  7701  prarloclemarch2  7780  prarloclemlt  7854  0cn  8312  resubcli  8583  0reALT  8617  10nn  9775  numsucc  9799  nummac  9804  qreccl  10025  unirnioo  10358  fz0to4untppr  10514  cats1fvn  11519  4sqlem19  13171  dec2dvds  13173  modsubi  13181  gcdi  13182  ballotfilemth  13264  fn0g  13678  fngzsum  13691  prdsex  14155  sn0topon  15172  retopbas  15607  blssioo  15637  hovercncf  15730  log2ublem2  16067  log2ublog2  16069  lgslem4  16105  konigsberglem1  16712  bj-unex  16928  exmidsbthrlem  17041