Theorem eleqtrri 2113

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 2044	. 2 ⊢ 𝐵 = 𝐶
4	1, 3	eleqtri 2112	1 ⊢ 𝐴 ∈ 𝐶

Syntax hints:   = wceq 1243   ∈ wcel 1393

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 99  ax-ia2 100  ax-ia3 101  ax-5 1336  ax-gen 1338  ax-ie1 1382  ax-ie2 1383  ax-4 1400  ax-17 1419  ax-ial 1427  ax-ext 2022

This theorem depends on definitions:  df-bi 110  df-cleq 2033  df-clel 2036

This theorem is referenced by:  3eltr4i  2119  opi1  3969  opi2  3970  ordpwsucexmid  4294  peano1  4317  acexmidlemcase  5507  acexmidlem2  5509  ac6sfi  6352  1lt2pi  6438  prarloclemarch2  6517  prarloclemlt  6591  prarloclemcalc  6600  pnfxr  8692  mnfxr  8694