Theorem eleqtrri 2265

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 2193	. 2 ⊢ 𝐵 = 𝐶
4	1, 3	eleqtri 2264	1 ⊢ 𝐴 ∈ 𝐶

Syntax hints:   = wceq 1364   ∈ wcel 2160

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 2171

This theorem depends on definitions:  df-bi 117  df-cleq 2182  df-clel 2185

This theorem is referenced by:  3eltr4i  2271  undifexmid  4208  opi1  4247  opi2  4248  ordpwsucexmid  4584  peano1  4608  acexmidlemcase  5887  acexmidlem2  5889  0lt2o  6461  1lt2o  6462  0elixp  6750  ac6sfi  6921  ctssdccl  7135  exmidomni  7165  exmidonfinlem  7217  exmidfodomrlemr  7226  exmidfodomrlemrALT  7227  exmidaclem  7232  pw1ne3  7254  3nelsucpw1  7258  1lt2pi  7364  prarloclemarch2  7443  prarloclemlt  7517  prarloclemcalc  7526  suplocexprlemdisj  7744  suplocexprlemub  7747  pnfxr  8035  mnfxr  8039  fnpr2ob  12809  dveflem  14624