Theorem 3eltr3i 2274

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

Hypotheses

Ref	Expression
3eltr3.1	|- A e. B
3eltr3.2	|- A = C
3eltr3.3	|- B = D

Assertion

Ref	Expression
3eltr3i	|- C e. D

Proof of Theorem 3eltr3i

Step	Hyp	Ref	Expression
1		3eltr3.2	. 2 |- A = C
2		3eltr3.1	. . 3 |- A e. B
3		3eltr3.3	. . 3 |- B = D
4	2, 3	eleqtri 2268	. 2 |- A e. D
5	1, 4	eqeltrri 2267	1 |- C e. D

Syntax hints:   = wceq 1364   e. wcel 2164

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 1458  ax-gen 1460  ax-ie1 1504  ax-ie2 1505  ax-4 1521  ax-17 1537  ax-ial 1545  ax-ext 2175

This theorem depends on definitions:  df-bi 117  df-cleq 2186  df-clel 2189

This theorem is referenced by: (None)