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Metamath Proof Explorer

Theorem ertr2d

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

Ref Expression
Hypotheses ersymb.1 
|- ( ph -> R Er X )
ertrd.5 
|- ( ph -> A R B )
ertrd.6 
|- ( ph -> B R C )
Assertion ertr2d 
|- ( ph -> C R A )

Proof

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