Metamath Proof Explorer

Theorem eqvrelcoss2

Description: Two ways to express equivalent cosets. (Contributed by Peter Mazsa, 3-May-2019)

Ref Expression
Assertion eqvrelcoss2 
|- ( EqvRel ,~ R <-> ,~ ,~ R C_ ,~ R )

Proof

Step Hyp Ref Expression
1 eqvrelcoss3 
 |-  ( EqvRel ,~ R <-> A. x A. y A. z ( ( x ,~ R y /\ y ,~ R z ) -> x ,~ R z ) )
2 cocossss 
 |-  ( ,~ ,~ R C_ ,~ R <-> A. x A. y A. z ( ( x ,~ R y /\ y ,~ R z ) -> x ,~ R z ) )
3 1 2 bitr4i 
 |-  ( EqvRel ,~ R <-> ,~ ,~ R C_ ,~ R )