Theorem cvbr4i

Description: An alternate way to express the covering property. (Contributed by NM, 30-Nov-2004) (New usage is discouraged.)

	Ref	Expression
Hypotheses	chpssat.1	|- A e. CH
	chpssat.2	|- B e. CH
Assertion	cvbr4i	|- ( A ( A C. B /\ E. x e. HAtoms ( A vH x ) = B ) )

Proof

Step	Hyp	Ref	Expression
1		chpssat.1	|- A e. CH
2		chpssat.2	|- B e. CH
3		cvpss	|- ( ( A e. CH /\ B e. CH ) -> ( A A C. B ) )
4	1 2 3	mp2an	|- ( A A C. B )
5	1 2	cvati	|- ( A E. x e. HAtoms ( A vH x ) = B )
6	4 5	jca	|- ( A ( A C. B /\ E. x e. HAtoms ( A vH x ) = B ) )
7		chcv2	|- ( ( A e. CH /\ x e. HAtoms ) -> ( A C. ( A vH x ) <-> A
8	1 7	mpan	|- ( x e. HAtoms -> ( A C. ( A vH x ) <-> A
9	8	adantr	|- ( ( x e. HAtoms /\ ( A vH x ) = B ) -> ( A C. ( A vH x ) <-> A
10		psseq2	|- ( ( A vH x ) = B -> ( A C. ( A vH x ) <-> A C. B ) )
11	10	adantl	|- ( ( x e. HAtoms /\ ( A vH x ) = B ) -> ( A C. ( A vH x ) <-> A C. B ) )
12		breq2	|- ( ( A vH x ) = B -> ( A A
13	12	adantl	|- ( ( x e. HAtoms /\ ( A vH x ) = B ) -> ( A A
14	9 11 13	3bitr3d	|- ( ( x e. HAtoms /\ ( A vH x ) = B ) -> ( A C. B <-> A
15	14	biimpd	|- ( ( x e. HAtoms /\ ( A vH x ) = B ) -> ( A C. B -> A
16	15	ex	|- ( x e. HAtoms -> ( ( A vH x ) = B -> ( A C. B -> A
17	16	com3r	|- ( A C. B -> ( x e. HAtoms -> ( ( A vH x ) = B -> A
18	17	rexlimdv	|- ( A C. B -> ( E. x e. HAtoms ( A vH x ) = B -> A
19	18	imp	|- ( ( A C. B /\ E. x e. HAtoms ( A vH x ) = B ) -> A
20	6 19	impbii	|- ( A ( A C. B /\ E. x e. HAtoms ( A vH x ) = B ) )