Theorem eqeltrri 2279

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 2209	. 2 ⊢ 𝐵 = 𝐴
3		eqeltrr.2	. 2 ⊢ 𝐴 ∈ 𝐶
4	2, 3	eqeltri 2278	1 ⊢ 𝐵 ∈ 𝐶

Syntax hints:   = wceq 1373   ∈ wcel 2176

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 1470  ax-gen 1472  ax-ie1 1516  ax-ie2 1517  ax-4 1533  ax-17 1549  ax-ial 1557  ax-ext 2187

This theorem depends on definitions:  df-bi 117  df-cleq 2198  df-clel 2201

This theorem is referenced by:  3eltr3i  2286  p0ex  4232  epse  4389  unex  4488  ordtri2orexmid  4571  onsucsssucexmid  4575  ordsoexmid  4610  ordtri2or2exmid  4619  ontri2orexmidim  4620  nnregexmid  4669  abrexex  6202  opabex3  6207  abrexex2  6209  abexssex  6210  abexex  6211  oprabrexex2  6215  tfr0dm  6408  exmidonfinlem  7301  1lt2pi  7453  prarloclemarch2  7532  prarloclemlt  7606  0cn  8064  resubcli  8335  0reALT  8369  10nn  9519  numsucc  9543  nummac  9548  qreccl  9763  unirnioo  10095  fz0to4untppr  10246  4sqlem19  12732  dec2dvds  12734  modsubi  12742  gcdi  12743  prdsex  13101  fn0g  13207  fngsum  13220  sn0topon  14560  retopbas  14995  blssioo  15025  hovercncf  15118  lgslem4  15480  bj-unex  15855  exmidsbthrlem  15961