Theorem eleqtrri 2314

Description: Substitution of equal classes into membership relation. (Contributed by NM, 5-Aug-1993.)

Hypotheses

Ref	Expression
eleqtrr.1	|- A e. B
eleqtrr.2	|- C = B

Assertion

Ref	Expression
eleqtrri	|- A e. C

Proof of Theorem eleqtrri

Step	Hyp	Ref	Expression
1		eleqtrr.1	. 2 |- A e. B
2		eleqtrr.2	. . 3 |- C = B
3	2	eqcomi 2242	. 2 |- B = C
4	1, 3	eleqtri 2313	1 |- A e. C

This proof depends on syntax axioms:   = wceq 1402   e. wcel 2209

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-5 1500  ax-gen 1502  ax-ie1 1546  ax-ie2 1547  ax-4 1563  ax-17 1579  ax-ial 1587  ax-ext 2220

This proof depends on definitions:  df-bi 117  df-cleq 2231  df-clel 2234

This theorem is used by:  3eltr4i  2320  undifexmid  4330  opi1  4372  opi2  4373  ordpwsucexmid  4717  peano1  4741  acexmidlemcase  6080  acexmidlem2  6082  0lt2o  6714  1lt2o  6715  0elixp  7011  ac6sfi  7202  ctssdccl  7452  exmidomni  7483  exmidonfinlem  7546  exmidfodomrlemr  7555  exmidfodomrlemrALT  7556  exmidaclem  7565  pw1ne3  7590  3nelsucpw1  7594  1lt2pi  7708  prarloclemarch2  7787  prarloclemlt  7861  prarloclemcalc  7870  suplocexprlemdisj  8088  suplocexprlemub  8091  pnfxr  8379  mnfxr  8383  0bits  12745  fnpr2ob  13714  dveflem  15918  konigsberglem4  16898  3dom  17184