Metamath Proof Explorer

Theorem relcnveq4

Description: Two ways of saying a relation is symmetric. (Contributed by Peter Mazsa, 28-Apr-2019)

Ref Expression
Assertion relcnveq4 
|- ( Rel R -> ( `' R C_ R <-> A. x A. y ( x R y <-> y R x ) ) )

Proof

Step Hyp Ref Expression
1 relcnveq 
 |-  ( Rel R -> ( `' R C_ R <-> `' R = R ) )
2 relcnveq2 
 |-  ( Rel R -> ( `' R = R <-> A. x A. y ( x R y <-> y R x ) ) )
3 1 2 bitrd 
 |-  ( Rel R -> ( `' R C_ R <-> A. x A. y ( x R y <-> y R x ) ) )