Theorem eqeltrri 2305

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 2235	. 2 ⊢ 𝐵 = 𝐴
3		eqeltrr.2	. 2 ⊢ 𝐴 ∈ 𝐶
4	2, 3	eqeltri 2304	1 ⊢ 𝐵 ∈ 𝐶

Syntax hints:   = wceq 1397   ∈ wcel 2202

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 1495  ax-gen 1497  ax-ie1 1541  ax-ie2 1542  ax-4 1558  ax-17 1574  ax-ial 1582  ax-ext 2213

This theorem depends on definitions:  df-bi 117  df-cleq 2224  df-clel 2227

This theorem is referenced by:  3eltr3i  2312  p0ex  4278  epse  4439  unex  4538  ordtri2orexmid  4621  onsucsssucexmid  4625  ordsoexmid  4660  ordtri2or2exmid  4669  ontri2orexmidim  4670  nnregexmid  4719  abrexex  6282  opabex3  6287  abrexex2  6289  abexssex  6290  abexex  6291  oprabrexex2  6295  tfr0dm  6491  exmidonfinlem  7407  1lt2pi  7563  prarloclemarch2  7642  prarloclemlt  7716  0cn  8174  resubcli  8445  0reALT  8479  10nn  9629  numsucc  9653  nummac  9658  qreccl  9879  unirnioo  10211  fz0to4untppr  10362  cats1fvn  11352  4sqlem19  13003  dec2dvds  13005  modsubi  13013  gcdi  13014  prdsex  13373  fn0g  13479  fngsum  13492  sn0topon  14839  retopbas  15274  blssioo  15304  hovercncf  15397  lgslem4  15759  konigsberglem1  16366  bj-unex  16573  exmidsbthrlem  16685