Theorem eleqtrri 2307

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 2235	. 2 |- B = C
4	1, 3	eleqtri 2306	1 |- A e. C

Syntax hints:   = wceq 1398   e. wcel 2202

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 1496  ax-gen 1498  ax-ie1 1542  ax-ie2 1543  ax-4 1559  ax-17 1575  ax-ial 1583  ax-ext 2213

This theorem depends on definitions:  df-bi 117  df-cleq 2224  df-clel 2227

This theorem is referenced by:  3eltr4i  2313  undifexmid  4289  opi1  4330  opi2  4331  ordpwsucexmid  4674  peano1  4698  acexmidlemcase  6023  acexmidlem2  6025  0lt2o  6652  1lt2o  6653  0elixp  6941  ac6sfi  7130  ctssdccl  7370  exmidomni  7401  exmidonfinlem  7464  exmidfodomrlemr  7473  exmidfodomrlemrALT  7474  exmidaclem  7483  pw1ne3  7508  3nelsucpw1  7512  1lt2pi  7620  prarloclemarch2  7699  prarloclemlt  7773  prarloclemcalc  7782  suplocexprlemdisj  8000  suplocexprlemub  8003  pnfxr  8291  mnfxr  8295  0bits  12600  fnpr2ob  13503  dveflem  15537  konigsberglem4  16432  3dom  16708