Theorem eleqtrri 2251

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 2179	. 2 ⊢ 𝐵 = 𝐶
4	1, 3	eleqtri 2250	1 ⊢ 𝐴 ∈ 𝐶

Syntax hints:   = wceq 1353   ∈ wcel 2146

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 1445  ax-gen 1447  ax-ie1 1491  ax-ie2 1492  ax-4 1508  ax-17 1524  ax-ial 1532  ax-ext 2157

This theorem depends on definitions:  df-bi 117  df-cleq 2168  df-clel 2171

This theorem is referenced by:  3eltr4i  2257  undifexmid  4188  opi1  4226  opi2  4227  ordpwsucexmid  4563  peano1  4587  acexmidlemcase  5860  acexmidlem2  5862  0lt2o  6432  1lt2o  6433  0elixp  6719  ac6sfi  6888  ctssdccl  7100  exmidomni  7130  exmidonfinlem  7182  exmidfodomrlemr  7191  exmidfodomrlemrALT  7192  exmidaclem  7197  pw1ne3  7219  3nelsucpw1  7223  1lt2pi  7314  prarloclemarch2  7393  prarloclemlt  7467  prarloclemcalc  7476  suplocexprlemdisj  7694  suplocexprlemub  7697  pnfxr  7984  mnfxr  7988  dveflem  13738