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Metamath Proof Explorer

Theorem ertr4d

Description: A transitivity relation for equivalences. (Contributed by Mario Carneiro, 9-Jul-2014)

Ref Expression
Hypotheses ersymb.1 
|- ( ph -> R Er X )
ertr4d.5 
|- ( ph -> A R B )
ertr4d.6 
|- ( ph -> C R B )
Assertion ertr4d 
|- ( ph -> A R C )

Proof

Step Hyp Ref Expression
1 ersymb.1 
 |-  ( ph -> R Er X )
2 ertr4d.5 
 |-  ( ph -> A R B )
3 ertr4d.6 
 |-  ( ph -> C R B )
4 1 3 ersym 
 |-  ( ph -> B R C )
5 1 2 4 ertrd 
 |-  ( ph -> A R C )