Theorem 3eltr3g 2251

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

Hypotheses

Ref	Expression
3eltr3g.1	|- ( ph -> A e. B )
3eltr3g.2	|- A = C
3eltr3g.3	|- B = D

Assertion

Ref	Expression
3eltr3g	|- ( ph -> C e. D )

Proof of Theorem 3eltr3g

Step	Hyp	Ref	Expression
1		3eltr3g.1	. 2 |- ( ph -> A e. B )
2		3eltr3g.2	. . 3 |- A = C
3		3eltr3g.3	. . 3 |- B = D
4	2, 3	eleq12i 2234	. 2 |- ( A e. B <-> C e. D )
5	1, 4	sylib 121	1 |- ( ph -> C e. D )

Syntax hints:   -> wi 4   = wceq 1343   e. wcel 2136

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-5 1435  ax-gen 1437  ax-ie1 1481  ax-ie2 1482  ax-4 1498  ax-17 1514  ax-ial 1522  ax-ext 2147

This theorem depends on definitions:  df-bi 116  df-cleq 2158  df-clel 2161

This theorem is referenced by: (None)