Theorem eleqtr 1161

Description: Substitution of equal classes into membership relation.

Hypotheses

Ref	Expression
eleqtr.1	⊢ A ∈ B
eleqtr.2	⊢ B = C

Assertion

Ref	Expression
eleqtr	⊢ A ∈ C

Proof of Theorem eleqtr

Step	Hyp	Ref	Expression
1		eleqtr.1	. 2 ⊢ A ∈ B
2		eleqtr.2	. . 3 ⊢ B = C
3	2	eleq2i 1153	. 2 ⊢ (A ∈ B ↔ A ∈ C)
4	1, 3	mpbi 164	1 ⊢ A ∈ C

Syntax hints:   = wceq 1091   ∈ wcel 1092

This theorem is referenced by:  eleqtrr 1162  pri2 1842  rankpw 3528

This theorem was proved from axioms:  ax-1 3  ax-2 4  ax-3 5  ax-mp 6  ax-4 673  ax-5 674  ax-gen 677  ax-17 925  ax-ext 1074

This theorem depends on definitions:  df-bi 128  df-an 198  df-ex 679  df-cleq 1097  df-clel 1099

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