Theorem eleqtri 2240

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

Hypotheses

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

Assertion

Ref	Expression
eleqtri	|- A e. C

Proof of Theorem eleqtri

Step	Hyp	Ref	Expression
1		eleqtr.1	. 2 |- A e. B
2		eleqtr.2	. . 3 |- B = C
3	2	eleq2i 2232	. 2 |- ( A e. B <-> A e. C )
4	1, 3	mpbi 144	1 |- A e. C

Syntax hints:   = wceq 1343   e. wcel 2136

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-5 1435  ax-gen 1437  ax-ie1 1481  ax-ie2 1482  ax-4 1498  ax-17 1514  ax-ial 1522  ax-ext 2147

This theorem depends on definitions:  df-bi 116  df-cleq 2158  df-clel 2161

This theorem is referenced by:  eleqtrri  2241  3eltr3i  2246  prid2  3682  2eluzge0  9509  fz01or  10042  fz0to4untppr  10055  ef0lem  11597  ege2le3  11608  efgt1p2  11632  efgt1p  11633  phi1  12147  cnrehmeocntop  13193  dvcjbr  13272  fmelpw1o  13648