Theorem eleqtrri 2314

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 2242	. 2 ⊢ 𝐵 = 𝐶
4	1, 3	eleqtri 2313	1 ⊢ 𝐴 ∈ 𝐶

This proof depends on syntax axioms:   = wceq 1402   ∈ wcel 2209

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-5 1500  ax-gen 1502  ax-ie1 1546  ax-ie2 1547  ax-4 1563  ax-17 1579  ax-ial 1587  ax-ext 2220

This proof depends on definitions:  df-bi 117  df-cleq 2231  df-clel 2234

This theorem is used by:  3eltr4i  2320  undifexmid  4332  opi1  4374  opi2  4375  ordpwsucexmid  4719  peano1  4743  acexmidlemcase  6083  acexmidlem2  6085  0lt2o  6718  1lt2o  6719  0elixp  7015  ac6sfi  7206  ctssdccl  7457  exmidomni  7488  exmidonfinlem  7551  exmidfodomrlemr  7560  exmidfodomrlemrALT  7561  exmidaclem  7570  pw1ne3  7595  3nelsucpw1  7599  1lt2pi  7713  prarloclemarch2  7792  prarloclemlt  7866  prarloclemcalc  7875  suplocexprlemdisj  8093  suplocexprlemub  8096  pnfxr  8384  mnfxr  8388  0bits  12751  fnpr2ob  13720  dveflem  15952  konigsberglem4  16937  3dom  17223