Theorem eleqtrri 2308

Description: Substitution of equal classes into membership relation. (Contributed by NM, 5-Aug-1993.)

Hypotheses

Ref	Expression
eleqtrr.1	⊢ 𝐴 ∈ 𝐵
eleqtrr.2	⊢ 𝐶 = 𝐵

Assertion

Ref	Expression
eleqtrri	⊢ 𝐴 ∈ 𝐶

Proof of Theorem eleqtrri

Step	Hyp	Ref	Expression
1		eleqtrr.1	. 2 ⊢ 𝐴 ∈ 𝐵
2		eleqtrr.2	. . 3 ⊢ 𝐶 = 𝐵
3	2	eqcomi 2236	. 2 ⊢ 𝐵 = 𝐶
4	1, 3	eleqtri 2307	1 ⊢ 𝐴 ∈ 𝐶

Syntax hints:   = wceq 1398   ∈ wcel 2203

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 1496  ax-gen 1498  ax-ie1 1542  ax-ie2 1543  ax-4 1559  ax-17 1575  ax-ial 1583  ax-ext 2214

This theorem depends on definitions:  df-bi 117  df-cleq 2225  df-clel 2228

This theorem is referenced by:  3eltr4i  2314  undifexmid  4306  opi1  4348  opi2  4349  ordpwsucexmid  4692  peano1  4716  acexmidlemcase  6045  acexmidlem2  6047  0lt2o  6674  1lt2o  6675  0elixp  6964  ac6sfi  7155  ctssdccl  7402  exmidomni  7433  exmidonfinlem  7496  exmidfodomrlemr  7505  exmidfodomrlemrALT  7506  exmidaclem  7515  pw1ne3  7540  3nelsucpw1  7544  1lt2pi  7655  prarloclemarch2  7734  prarloclemlt  7808  prarloclemcalc  7817  suplocexprlemdisj  8035  suplocexprlemub  8038  pnfxr  8326  mnfxr  8330  0bits  12645  fnpr2ob  13553  dveflem  15591  konigsberglem4  16486  3dom  16762