Theorem 3eltr4g 2263

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

Hypotheses

Ref	Expression
3eltr4g.1	|- ( ph -> A e. B )
3eltr4g.2	|- C = A
3eltr4g.3	|- D = B

Assertion

Ref	Expression
3eltr4g	|- ( ph -> C e. D )

Proof of Theorem 3eltr4g

Step	Hyp	Ref	Expression
1		3eltr4g.1	. 2 |- ( ph -> A e. B )
2		3eltr4g.2	. . 3 |- C = A
3		3eltr4g.3	. . 3 |- D = B
4	2, 3	eleq12i 2245	. 2 |- ( C e. D <-> A e. B )
5	1, 4	sylibr 134	1 |- ( ph -> C e. D )

Syntax hints:   -> wi 4   = wceq 1353   e. wcel 2148

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 1447  ax-gen 1449  ax-ie1 1493  ax-ie2 1494  ax-4 1510  ax-17 1526  ax-ial 1534  ax-ext 2159

This theorem depends on definitions:  df-bi 117  df-cleq 2170  df-clel 2173

This theorem is referenced by:  riotacl2  5840  2strop1g  12573