Theorem eleqtri 2307

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 2299	. 2 |- ( A e. B <-> A e. C )
4	1, 3	mpbi 145	1 |- A e. C

Syntax hints:   = wceq 1398   e. wcel 2203

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 1496  ax-gen 1498  ax-ie1 1542  ax-ie2 1543  ax-4 1559  ax-17 1575  ax-ial 1583  ax-ext 2214

This theorem depends on definitions:  df-bi 117  df-cleq 2225  df-clel 2228

This theorem is referenced by:  eleqtrri  2308  3eltr3i  2313  prid2  3797  2eluzge0  9903  fz01or  10441  fz0to4untppr  10454  ef0lem  12339  ege2le3  12350  efgt1p2  12374  efgt1p  12375  phi1  12909  cnrehmeocntop  15462  dvcjbr  15560