Theorem eleqtri 2169

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 2161	. 2 ⊢ (𝐴 ∈ 𝐵 ↔ 𝐴 ∈ 𝐶)
4	1, 3	mpbi 144	1 ⊢ 𝐴 ∈ 𝐶

Syntax hints:   = wceq 1296   ∈ wcel 1445

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-5 1388  ax-gen 1390  ax-ie1 1434  ax-ie2 1435  ax-4 1452  ax-17 1471  ax-ial 1479  ax-ext 2077

This theorem depends on definitions:  df-bi 116  df-cleq 2088  df-clel 2091

This theorem is referenced by:  eleqtrri  2170  3eltr3i  2175  prid2  3569  2eluzge0  9162  fz01or  9674  ef0lem  11099  ege2le3  11110  efgt1p2  11134  efgt1p  11135  phi1  11622