Theorem eleqtri 2279

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 2271	. 2 ⊢ (𝐴 ∈ 𝐵 ↔ 𝐴 ∈ 𝐶)
4	1, 3	mpbi 145	1 ⊢ 𝐴 ∈ 𝐶

Syntax hints:   = wceq 1372   ∈ wcel 2175

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 1469  ax-gen 1471  ax-ie1 1515  ax-ie2 1516  ax-4 1532  ax-17 1548  ax-ial 1556  ax-ext 2186

This theorem depends on definitions:  df-bi 117  df-cleq 2197  df-clel 2200

This theorem is referenced by:  eleqtrri  2280  3eltr3i  2285  prid2  3739  2eluzge0  9695  fz01or  10232  fz0to4untppr  10245  ef0lem  11913  ege2le3  11924  efgt1p2  11948  efgt1p  11949  phi1  12483  cnrehmeocntop  15024  dvcjbr  15122  fmelpw1o  15675