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  4221  epse  4377  unex  4476  ordtri2orexmid  4559  onsucsssucexmid  4563  ordsoexmid  4598  ordtri2or2exmid  4607  ontri2orexmidim  4608  nnregexmid  4657  abrexex  6174  opabex3  6179  abrexex2  6181  abexssex  6182  abexex  6183  oprabrexex2  6187  tfr0dm  6380  exmidonfinlem  7260  1lt2pi  7407  prarloclemarch2  7486  prarloclemlt  7560  0cn  8018  resubcli  8289  0reALT  8323  10nn  9472  numsucc  9496  nummac  9501  qreccl  9716  unirnioo  10048  fz0to4untppr  10199  4sqlem19  12578  dec2dvds  12580  modsubi  12588  gcdi  12589  prdsex  12940  fn0g  13018  fngsum  13031  sn0topon  14324  retopbas  14759  blssioo  14789  hovercncf  14882  lgslem4  15244  bj-unex  15565  exmidsbthrlem  15666