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Metamath Proof Explorer

Theorem breqtrd

Description: Substitution of equal classes into a binary relation. (Contributed by NM, 24-Oct-1999)

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

Proof

Step Hyp Ref Expression
1 breqtrd.1 
 |-  ( ph -> A R B )
2 breqtrd.2 
 |-  ( ph -> B = C )
3 2 breq2d 
 |-  ( ph -> ( A R B <-> A R C ) )
4 1 3 mpbid 
 |-  ( ph -> A R C )