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Metamath Proof Explorer

Theorem eqeltrrd

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

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

Proof

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