Theorem eleqtri 2264

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 2256	. 2 ⊢ (𝐴 ∈ 𝐵 ↔ 𝐴 ∈ 𝐶)
4	1, 3	mpbi 145	1 ⊢ 𝐴 ∈ 𝐶

Syntax hints:   = wceq 1364   ∈ wcel 2160

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 2171

This theorem depends on definitions:  df-bi 117  df-cleq 2182  df-clel 2185

This theorem is referenced by:  eleqtrri  2265  3eltr3i  2270  prid2  3714  2eluzge0  9605  fz01or  10141  fz0to4untppr  10154  ef0lem  11700  ege2le3  11711  efgt1p2  11735  efgt1p  11736  phi1  12251  cnrehmeocntop  14553  dvcjbr  14632  fmelpw1o  15019