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Metamath Proof Explorer

Theorem 3eltr3d

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

Ref Expression
Hypotheses 3eltr3d.1 
|- ( ph -> A e. B )
3eltr3d.2 
|- ( ph -> A = C )
3eltr3d.3 
|- ( ph -> B = D )
Assertion 3eltr3d 
|- ( ph -> C e. D )

Proof

Step Hyp Ref Expression
1 3eltr3d.1 
 |-  ( ph -> A e. B )
2 3eltr3d.2 
 |-  ( ph -> A = C )
3 3eltr3d.3 
 |-  ( ph -> B = D )
4 1 3 eleqtrd 
 |-  ( ph -> A e. D )
5 2 4 eqeltrrd 
 |-  ( ph -> C e. D )