Theorem chcon3i

Description: Hilbert lattice contraposition law. (Contributed by NM, 24-Jun-2004) (New usage is discouraged.)

	Ref	Expression
Hypotheses	ch0le.1	|- A e. CH
	chjcl.2	|- B e. CH
Assertion	chcon3i	|- ( A = B <-> ( _|_ ` B ) = ( _|_ ` A ) )

Proof

Step	Hyp	Ref	Expression
1		ch0le.1	|- A e. CH
2		chjcl.2	|- B e. CH
3	1 2	chsscon3i	|- ( A C_ B <-> ( _|_ ` B ) C_ ( _|_ ` A ) )
4	2 1	chsscon3i	|- ( B C_ A <-> ( _|_ ` A ) C_ ( _|_ ` B ) )
5	3 4	anbi12i	|- ( ( A C_ B /\ B C_ A ) <-> ( ( _|_ ` B ) C_ ( _|_ ` A ) /\ ( _|_ ` A ) C_ ( _|_ ` B ) ) )
6		eqss	|- ( A = B <-> ( A C_ B /\ B C_ A ) )
7		eqss	|- ( ( _|_ ` B ) = ( _|_ ` A ) <-> ( ( _|_ ` B ) C_ ( _|_ ` A ) /\ ( _|_ ` A ) C_ ( _|_ ` B ) ) )
8	5 6 7	3bitr4i	|- ( A = B <-> ( _|_ ` B ) = ( _|_ ` A ) )