Theorem 3eltr3i 2431

Description: Substitution of equal classes into membership relation. (Contributed by Mario Carneiro, 6-Jan-2017.)

Hypotheses

Ref	Expression
3eltr3.1	⊢ A ∈ B
3eltr3.2	⊢ A = C
3eltr3.3	⊢ B = D

Assertion

Ref	Expression
3eltr3i	⊢ C ∈ D

Proof of Theorem 3eltr3i

Step	Hyp	Ref	Expression
1		3eltr3.2	. 2 ⊢ A = C
2		3eltr3.1	. . 3 ⊢ A ∈ B
3		3eltr3.3	. . 3 ⊢ B = D
4	2, 3	eleqtri 2425	. 2 ⊢ A ∈ D
5	1, 4	eqeltrri 2424	1 ⊢ C ∈ D

Syntax hints:   = wceq 1642   ∈ wcel 1710

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1546  ax-5 1557  ax-17 1616  ax-9 1654  ax-8 1675  ax-11 1746  ax-ext 2334

This theorem depends on definitions:  df-bi 177  df-an 360  df-ex 1542  df-cleq 2346  df-clel 2349

This theorem is referenced by: (None)