Theorem 3eltr4i 2287

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

Hypotheses

Ref	Expression
3eltr4.1	|- A e. B
3eltr4.2	|- C = A
3eltr4.3	|- D = B

Assertion

Ref	Expression
3eltr4i	|- C e. D

Proof of Theorem 3eltr4i

Step	Hyp	Ref	Expression
1		3eltr4.2	. 2 |- C = A
2		3eltr4.1	. . 3 |- A e. B
3		3eltr4.3	. . 3 |- D = B
4	2, 3	eleqtrri 2281	. 2 |- A e. D
5	1, 4	eqeltri 2278	1 |- C e. D

Syntax hints:   = wceq 1373   e. wcel 2176

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 1470  ax-gen 1472  ax-ie1 1516  ax-ie2 1517  ax-4 1533  ax-17 1549  ax-ial 1557  ax-ext 2187

This theorem depends on definitions:  df-bi 117  df-cleq 2198  df-clel 2201

This theorem is referenced by:  1nq  7481  0r  7865  1sr  7866  m1r  7867