Theorem eleqtrri 2307

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 2235	. 2 ⊢ 𝐵 = 𝐶
4	1, 3	eleqtri 2306	1 ⊢ 𝐴 ∈ 𝐶

Syntax hints:   = wceq 1398   ∈ wcel 2202

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 1496  ax-gen 1498  ax-ie1 1542  ax-ie2 1543  ax-4 1559  ax-17 1575  ax-ial 1583  ax-ext 2213

This theorem depends on definitions:  df-bi 117  df-cleq 2224  df-clel 2227

This theorem is referenced by:  3eltr4i  2313  undifexmid  4289  opi1  4330  opi2  4331  ordpwsucexmid  4674  peano1  4698  acexmidlemcase  6023  acexmidlem2  6025  0lt2o  6652  1lt2o  6653  0elixp  6941  ac6sfi  7130  ctssdccl  7353  exmidomni  7384  exmidonfinlem  7447  exmidfodomrlemr  7456  exmidfodomrlemrALT  7457  exmidaclem  7466  pw1ne3  7491  3nelsucpw1  7495  1lt2pi  7603  prarloclemarch2  7682  prarloclemlt  7756  prarloclemcalc  7765  suplocexprlemdisj  7983  suplocexprlemub  7986  pnfxr  8274  mnfxr  8278  0bits  12583  fnpr2ob  13486  dveflem  15520  konigsberglem4  16415  3dom  16691