Theorem dfsymrel3

Description: Alternate definition of the symmetric relation predicate. (Contributed by Peter Mazsa, 21-Apr-2019) (Revised by Peter Mazsa, 17-Aug-2021)

	Ref	Expression
Assertion	dfsymrel3	|- ( SymRel R <-> ( A. x A. y ( x R y -> y R x ) /\ Rel R ) )

Proof

Step	Hyp	Ref	Expression
1		dfsymrel2	|- ( SymRel R <-> ( `' R C_ R /\ Rel R ) )
2		cnvsym	|- ( `' R C_ R <-> A. x A. y ( x R y -> y R x ) )
3	2	anbi1i	|- ( ( `' R C_ R /\ Rel R ) <-> ( A. x A. y ( x R y -> y R x ) /\ Rel R ) )
4	1 3	bitri	|- ( SymRel R <-> ( A. x A. y ( x R y -> y R x ) /\ Rel R ) )