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  4281  opi1  4322  opi2  4323  ordpwsucexmid  4666  peano1  4690  acexmidlemcase  6008  acexmidlem2  6010  0lt2o  6604  1lt2o  6605  0elixp  6893  ac6sfi  7080  ctssdccl  7301  exmidomni  7332  exmidonfinlem  7394  exmidfodomrlemr  7403  exmidfodomrlemrALT  7404  exmidaclem  7413  pw1ne3  7438  3nelsucpw1  7442  1lt2pi  7550  prarloclemarch2  7629  prarloclemlt  7703  prarloclemcalc  7712  suplocexprlemdisj  7930  suplocexprlemub  7933  pnfxr  8222  mnfxr  8226  0bits  12510  fnpr2ob  13413  dveflem  15440  3dom  16523