Theorem eleqtri 2313

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 2305	. 2 ⊢ (𝐴 ∈ 𝐵 ↔ 𝐴 ∈ 𝐶)
4	1, 3	mpbi 145	1 ⊢ 𝐴 ∈ 𝐶

Syntax hints:   = wceq 1402   ∈ wcel 2209

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 1500  ax-gen 1502  ax-ie1 1546  ax-ie2 1547  ax-4 1563  ax-17 1579  ax-ial 1587  ax-ext 2220

This theorem depends on definitions:  df-bi 117  df-cleq 2231  df-clel 2234

This theorem is referenced by:  eleqtrri  2314  3eltr3i  2319  prid2  3814  2eluzge0  9954  fz01or  10496  fz0to4untppr  10509  ef0lem  12405  ege2le3  12416  efgt1p2  12440  efgt1p  12441  phi1  12975  ballotfilem2  13206  ballotfilem1ri  13256  cnrehmeocntop  15634  dvcjbr  15732