Theorem eleqtri 2304

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 2296	. 2 |- ( A e. B <-> A e. C )
4	1, 3	mpbi 145	1 |- A e. C

Syntax hints:   = wceq 1395   e. wcel 2200

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 1493  ax-gen 1495  ax-ie1 1539  ax-ie2 1540  ax-4 1556  ax-17 1572  ax-ial 1580  ax-ext 2211

This theorem depends on definitions:  df-bi 117  df-cleq 2222  df-clel 2225

This theorem is referenced by:  eleqtrri  2305  3eltr3i  2310  prid2  3773  fmelpw1o  7432  2eluzge0  9770  fz01or  10307  fz0to4untppr  10320  ef0lem  12171  ege2le3  12182  efgt1p2  12206  efgt1p  12207  phi1  12741  cnrehmeocntop  15284  dvcjbr  15382