Theorem eleqtrri 2269

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 2197	. 2 ⊢ 𝐵 = 𝐶
4	1, 3	eleqtri 2268	1 ⊢ 𝐴 ∈ 𝐶

Syntax hints:   = wceq 1364   ∈ wcel 2164

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 1458  ax-gen 1460  ax-ie1 1504  ax-ie2 1505  ax-4 1521  ax-17 1537  ax-ial 1545  ax-ext 2175

This theorem depends on definitions:  df-bi 117  df-cleq 2186  df-clel 2189

This theorem is referenced by:  3eltr4i  2275  undifexmid  4223  opi1  4262  opi2  4263  ordpwsucexmid  4603  peano1  4627  acexmidlemcase  5914  acexmidlem2  5916  0lt2o  6496  1lt2o  6497  0elixp  6785  ac6sfi  6956  ctssdccl  7172  exmidomni  7203  exmidonfinlem  7255  exmidfodomrlemr  7264  exmidfodomrlemrALT  7265  exmidaclem  7270  pw1ne3  7292  3nelsucpw1  7296  1lt2pi  7402  prarloclemarch2  7481  prarloclemlt  7555  prarloclemcalc  7564  suplocexprlemdisj  7782  suplocexprlemub  7785  pnfxr  8074  mnfxr  8078  fnpr2ob  12926  dveflem  14905