Theorem chpsscon2

Description: Hilbert lattice contraposition law for strict ordering. (Contributed by NM, 12-Jun-2004) (New usage is discouraged.)

	Ref	Expression
Assertion	chpsscon2	|- ( ( A e. CH /\ B e. CH ) -> ( A C. ( _|_ ` B ) <-> B C. ( _|_ ` A ) ) )

Proof

Step	Hyp	Ref	Expression
1		choccl	|- ( B e. CH -> ( _|_ ` B ) e. CH )
2		chpsscon3	|- ( ( A e. CH /\ ( _|_ ` B ) e. CH ) -> ( A C. ( _|_ ` B ) <-> ( _|_ ` ( _|_ ` B ) ) C. ( _|_ ` A ) ) )
3	1 2	sylan2	|- ( ( A e. CH /\ B e. CH ) -> ( A C. ( _|_ ` B ) <-> ( _|_ ` ( _|_ ` B ) ) C. ( _|_ ` A ) ) )
4		ococ	|- ( B e. CH -> ( _|_ ` ( _|_ ` B ) ) = B )
5	4	adantl	|- ( ( A e. CH /\ B e. CH ) -> ( _|_ ` ( _|_ ` B ) ) = B )
6	5	psseq1d	|- ( ( A e. CH /\ B e. CH ) -> ( ( _|_ ` ( _|_ ` B ) ) C. ( _|_ ` A ) <-> B C. ( _|_ ` A ) ) )
7	3 6	bitrd	|- ( ( A e. CH /\ B e. CH ) -> ( A C. ( _|_ ` B ) <-> B C. ( _|_ ` A ) ) )