Theorem chlejb2

Description: Hilbert lattice ordering in terms of join. (Contributed by NM, 2-Jul-2004) (New usage is discouraged.)

	Ref	Expression
Assertion	chlejb2	|- ( ( A e. CH /\ B e. CH ) -> ( A C_ B <-> ( B vH A ) = B ) )

Proof

Step	Hyp	Ref	Expression
1		chlejb1	|- ( ( A e. CH /\ B e. CH ) -> ( A C_ B <-> ( A vH B ) = B ) )
2		chjcom	|- ( ( A e. CH /\ B e. CH ) -> ( A vH B ) = ( B vH A ) )
3	2	eqeq1d	|- ( ( A e. CH /\ B e. CH ) -> ( ( A vH B ) = B <-> ( B vH A ) = B ) )
4	1 3	bitrd	|- ( ( A e. CH /\ B e. CH ) -> ( A C_ B <-> ( B vH A ) = B ) )