Theorem eleqtrri 2154

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 2085	. 2 |- B = C
4	1, 3	eleqtri 2153	1 |- A e. C

Syntax hints:   = wceq 1284   e. wcel 1433

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106  ax-5 1376  ax-gen 1378  ax-ie1 1422  ax-ie2 1423  ax-4 1440  ax-17 1459  ax-ial 1467  ax-ext 2063

This theorem depends on definitions:  df-bi 115  df-cleq 2074  df-clel 2077

This theorem is referenced by:  3eltr4i  2160  opi1  3987  opi2  3988  ordpwsucexmid  4313  peano1  4335  acexmidlemcase  5527  acexmidlem2  5529  ac6sfi  6379  1lt2pi  6530  prarloclemarch2  6609  prarloclemlt  6683  prarloclemcalc  6692  pnfxr  8846  mnfxr  8848