Theorem strb

Description: Strong state theorem (bidirectional version). (Contributed by NM, 7-Apr-2001) (New usage is discouraged.)

	Ref	Expression
Hypotheses	str.1	|- A e. CH
	str.2	|- B e. CH
Assertion	strb	|- ( A. f e. States ( ( f ` A ) = 1 -> ( f ` B ) = 1 ) <-> A C_ B )

Proof

Step	Hyp	Ref	Expression
1		str.1	|- A e. CH
2		str.2	|- B e. CH
3	1 2	stri	|- ( A. f e. States ( ( f ` A ) = 1 -> ( f ` B ) = 1 ) -> A C_ B )
4	1 2	stlesi	|- ( f e. States -> ( A C_ B -> ( ( f ` A ) = 1 -> ( f ` B ) = 1 ) ) )
5	4	com12	|- ( A C_ B -> ( f e. States -> ( ( f ` A ) = 1 -> ( f ` B ) = 1 ) ) )
6	5	ralrimiv	|- ( A C_ B -> A. f e. States ( ( f ` A ) = 1 -> ( f ` B ) = 1 ) )
7	3 6	impbii	|- ( A. f e. States ( ( f ` A ) = 1 -> ( f ` B ) = 1 ) <-> A C_ B )