Theorem eqeltrri 2270

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 2200	. 2 ⊢ 𝐵 = 𝐴
3		eqeltrr.2	. 2 ⊢ 𝐴 ∈ 𝐶
4	2, 3	eqeltri 2269	1 ⊢ 𝐵 ∈ 𝐶

Syntax hints:   = wceq 1364   ∈ wcel 2167

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 1461  ax-gen 1463  ax-ie1 1507  ax-ie2 1508  ax-4 1524  ax-17 1540  ax-ial 1548  ax-ext 2178

This theorem depends on definitions:  df-bi 117  df-cleq 2189  df-clel 2192

This theorem is referenced by:  3eltr3i  2277  p0ex  4222  epse  4378  unex  4477  ordtri2orexmid  4560  onsucsssucexmid  4564  ordsoexmid  4599  ordtri2or2exmid  4608  ontri2orexmidim  4609  nnregexmid  4658  abrexex  6183  opabex3  6188  abrexex2  6190  abexssex  6191  abexex  6192  oprabrexex2  6196  tfr0dm  6389  exmidonfinlem  7272  1lt2pi  7424  prarloclemarch2  7503  prarloclemlt  7577  0cn  8035  resubcli  8306  0reALT  8340  10nn  9489  numsucc  9513  nummac  9518  qreccl  9733  unirnioo  10065  fz0to4untppr  10216  4sqlem19  12603  dec2dvds  12605  modsubi  12613  gcdi  12614  prdsex  12971  fn0g  13077  fngsum  13090  sn0topon  14408  retopbas  14843  blssioo  14873  hovercncf  14966  lgslem4  15328  bj-unex  15649  exmidsbthrlem  15753