Theorem eleqtri 2268

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 2260	. 2 |- ( A e. B <-> A e. C )
4	1, 3	mpbi 145	1 |- A e. C

Syntax hints:   = wceq 1364   e. wcel 2164

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 1458  ax-gen 1460  ax-ie1 1504  ax-ie2 1505  ax-4 1521  ax-17 1537  ax-ial 1545  ax-ext 2175

This theorem depends on definitions:  df-bi 117  df-cleq 2186  df-clel 2189

This theorem is referenced by:  eleqtrri  2269  3eltr3i  2274  prid2  3725  2eluzge0  9640  fz01or  10177  fz0to4untppr  10190  ef0lem  11803  ege2le3  11814  efgt1p2  11838  efgt1p  11839  phi1  12357  cnrehmeocntop  14764  dvcjbr  14857  fmelpw1o  15298