Theorem eleqtrri 2272

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 2200	. 2 ⊢ 𝐵 = 𝐶
4	1, 3	eleqtri 2271	1 ⊢ 𝐴 ∈ 𝐶

Syntax hints:   = wceq 1364   ∈ wcel 2167

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 1461  ax-gen 1463  ax-ie1 1507  ax-ie2 1508  ax-4 1524  ax-17 1540  ax-ial 1548  ax-ext 2178

This theorem depends on definitions:  df-bi 117  df-cleq 2189  df-clel 2192

This theorem is referenced by:  3eltr4i  2278  undifexmid  4227  opi1  4266  opi2  4267  ordpwsucexmid  4607  peano1  4631  acexmidlemcase  5920  acexmidlem2  5922  0lt2o  6508  1lt2o  6509  0elixp  6797  ac6sfi  6968  ctssdccl  7186  exmidomni  7217  exmidonfinlem  7272  exmidfodomrlemr  7281  exmidfodomrlemrALT  7282  exmidaclem  7291  pw1ne3  7313  3nelsucpw1  7317  1lt2pi  7424  prarloclemarch2  7503  prarloclemlt  7577  prarloclemcalc  7586  suplocexprlemdisj  7804  suplocexprlemub  7807  pnfxr  8096  mnfxr  8100  0bits  12141  fnpr2ob  13042  dveflem  15046