Theorem eleqtri 2306

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 2298	. 2 |- ( A e. B <-> A e. C )
4	1, 3	mpbi 145	1 |- A e. C

Syntax hints:   = wceq 1397   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 1495  ax-gen 1497  ax-ie1 1541  ax-ie2 1542  ax-4 1558  ax-17 1574  ax-ial 1582  ax-ext 2213

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

This theorem is referenced by:  eleqtrri  2307  3eltr3i  2312  prid2  3778  fmelpw1o  7465  2eluzge0  9809  fz01or  10346  fz0to4untppr  10359  ef0lem  12223  ege2le3  12234  efgt1p2  12258  efgt1p  12259  phi1  12793  cnrehmeocntop  15337  dvcjbr  15435