Theorem eleqtrri 2281

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 2209	. 2 ⊢ 𝐵 = 𝐶
4	1, 3	eleqtri 2280	1 ⊢ 𝐴 ∈ 𝐶

Syntax hints:   = wceq 1373   ∈ wcel 2176

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 1470  ax-gen 1472  ax-ie1 1516  ax-ie2 1517  ax-4 1533  ax-17 1549  ax-ial 1557  ax-ext 2187

This theorem depends on definitions:  df-bi 117  df-cleq 2198  df-clel 2201

This theorem is referenced by:  3eltr4i  2287  undifexmid  4237  opi1  4276  opi2  4277  ordpwsucexmid  4618  peano1  4642  acexmidlemcase  5939  acexmidlem2  5941  0lt2o  6527  1lt2o  6528  0elixp  6816  ac6sfi  6995  ctssdccl  7213  exmidomni  7244  exmidonfinlem  7301  exmidfodomrlemr  7310  exmidfodomrlemrALT  7311  exmidaclem  7320  pw1ne3  7342  3nelsucpw1  7346  1lt2pi  7453  prarloclemarch2  7532  prarloclemlt  7606  prarloclemcalc  7615  suplocexprlemdisj  7833  suplocexprlemub  7836  pnfxr  8125  mnfxr  8129  0bits  12270  fnpr2ob  13172  dveflem  15198