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  4222  opi1  4261  opi2  4262  ordpwsucexmid  4602  peano1  4626  acexmidlemcase  5913  acexmidlem2  5915  0lt2o  6494  1lt2o  6495  0elixp  6783  ac6sfi  6954  ctssdccl  7170  exmidomni  7201  exmidonfinlem  7253  exmidfodomrlemr  7262  exmidfodomrlemrALT  7263  exmidaclem  7268  pw1ne3  7290  3nelsucpw1  7294  1lt2pi  7400  prarloclemarch2  7479  prarloclemlt  7553  prarloclemcalc  7562  suplocexprlemdisj  7780  suplocexprlemub  7783  pnfxr  8072  mnfxr  8076  fnpr2ob  12923  dveflem  14872