Theorem eleqtrri 2305

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 2233	. 2 ⊢ 𝐵 = 𝐶
4	1, 3	eleqtri 2304	1 ⊢ 𝐴 ∈ 𝐶

Syntax hints:   = wceq 1395   ∈ wcel 2200

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 1493  ax-gen 1495  ax-ie1 1539  ax-ie2 1540  ax-4 1556  ax-17 1572  ax-ial 1580  ax-ext 2211

This theorem depends on definitions:  df-bi 117  df-cleq 2222  df-clel 2225

This theorem is referenced by:  3eltr4i  2311  undifexmid  4276  opi1  4317  opi2  4318  ordpwsucexmid  4661  peano1  4685  acexmidlemcase  5995  acexmidlem2  5997  0lt2o  6585  1lt2o  6586  0elixp  6874  ac6sfi  7056  ctssdccl  7274  exmidomni  7305  exmidonfinlem  7367  exmidfodomrlemr  7376  exmidfodomrlemrALT  7377  exmidaclem  7386  pw1ne3  7411  3nelsucpw1  7415  1lt2pi  7523  prarloclemarch2  7602  prarloclemlt  7676  prarloclemcalc  7685  suplocexprlemdisj  7903  suplocexprlemub  7906  pnfxr  8195  mnfxr  8199  0bits  12465  fnpr2ob  13368  dveflem  15394