Theorem eleqtri 2245

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 2237	. 2 ⊢ (𝐴 ∈ 𝐵 ↔ 𝐴 ∈ 𝐶)
4	1, 3	mpbi 144	1 ⊢ 𝐴 ∈ 𝐶

Syntax hints:   = wceq 1348   ∈ wcel 2141

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-5 1440  ax-gen 1442  ax-ie1 1486  ax-ie2 1487  ax-4 1503  ax-17 1519  ax-ial 1527  ax-ext 2152

This theorem depends on definitions:  df-bi 116  df-cleq 2163  df-clel 2166

This theorem is referenced by:  eleqtrri  2246  3eltr3i  2251  prid2  3690  2eluzge0  9534  fz01or  10067  fz0to4untppr  10080  ef0lem  11623  ege2le3  11634  efgt1p2  11658  efgt1p  11659  phi1  12173  cnrehmeocntop  13387  dvcjbr  13466  fmelpw1o  13841