Theorem eleqtri 2425

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

Hypotheses

Ref	Expression
eleqtr.1	⊢ A ∈ B
eleqtr.2	⊢ B = C

Assertion

Ref	Expression
eleqtri	⊢ A ∈ C

Proof of Theorem eleqtri

Step	Hyp	Ref	Expression
1		eleqtr.1	. 2 ⊢ A ∈ B
2		eleqtr.2	. . 3 ⊢ B = C
3	2	eleq2i 2417	. 2 ⊢ (A ∈ B ↔ A ∈ C)
4	1, 3	mpbi 199	1 ⊢ A ∈ C

Syntax hints:   = wceq 1642   ∈ wcel 1710

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1546  ax-5 1557  ax-17 1616  ax-9 1654  ax-8 1675  ax-11 1746  ax-ext 2334

This theorem depends on definitions:  df-bi 177  df-an 360  df-ex 1542  df-cleq 2346  df-clel 2349

This theorem is referenced by:  eleqtrri  2426  3eltr3i  2431  prid2  3829