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  4226  opi1  4265  opi2  4266  ordpwsucexmid  4606  peano1  4630  acexmidlemcase  5917  acexmidlem2  5919  0lt2o  6499  1lt2o  6500  0elixp  6788  ac6sfi  6959  ctssdccl  7177  exmidomni  7208  exmidonfinlem  7260  exmidfodomrlemr  7269  exmidfodomrlemrALT  7270  exmidaclem  7275  pw1ne3  7297  3nelsucpw1  7301  1lt2pi  7407  prarloclemarch2  7486  prarloclemlt  7560  prarloclemcalc  7569  suplocexprlemdisj  7787  suplocexprlemub  7790  pnfxr  8079  mnfxr  8083  fnpr2ob  12983  dveflem  14962