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Metamath Proof Explorer

Theorem eleqtrrd

Description: Deduction that substitutes equal classes into membership. (Contributed by NM, 14-Dec-2004)

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

Proof

Step Hyp Ref Expression
1 eleqtrrd.1 
 |-  ( ph -> A e. B )
2 eleqtrrd.2 
 |-  ( ph -> C = B )
3 2 eqcomd 
 |-  ( ph -> B = C )
4 1 3 eleqtrd 
 |-  ( ph -> A e. C )