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  4277  opi1  4318  opi2  4319  ordpwsucexmid  4662  peano1  4686  acexmidlemcase  6002  acexmidlem2  6004  0lt2o  6595  1lt2o  6596  0elixp  6884  ac6sfi  7068  ctssdccl  7289  exmidomni  7320  exmidonfinlem  7382  exmidfodomrlemr  7391  exmidfodomrlemrALT  7392  exmidaclem  7401  pw1ne3  7426  3nelsucpw1  7430  1lt2pi  7538  prarloclemarch2  7617  prarloclemlt  7691  prarloclemcalc  7700  suplocexprlemdisj  7918  suplocexprlemub  7921  pnfxr  8210  mnfxr  8214  0bits  12485  fnpr2ob  13388  dveflem  15415  3dom  16411