Theorem chcon2i

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	chcon2i	|- ( A = ( _|_ ` B ) <-> B = ( _|_ ` A ) )

Proof

Step	Hyp	Ref	Expression
1		ch0le.1	|- A e. CH
2		chjcl.2	|- B e. CH
3	1 2	chsscon2i	|- ( A C_ ( _|_ ` B ) <-> B C_ ( _|_ ` A ) )
4	2 1	chsscon1i	|- ( ( _|_ ` 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 ) )