Theorem dfbi

Description: Definition df-bi rewritten in an abbreviated form to help intuitive understanding of that definition. Note that it is a conjunction of two implications; one which asserts properties that follow from the biconditional and one which asserts properties that imply the biconditional. (Contributed by NM, 15-Aug-2008)

	Ref	Expression
Assertion	dfbi	|- ( ( ( ph <-> ps ) -> ( ( ph -> ps ) /\ ( ps -> ph ) ) ) /\ ( ( ( ph -> ps ) /\ ( ps -> ph ) ) -> ( ph <-> ps ) ) )

Proof

Step	Hyp	Ref	Expression
1		dfbi2	|- ( ( ph <-> ps ) <-> ( ( ph -> ps ) /\ ( ps -> ph ) ) )
2		dfbi2	|- ( ( ( ph <-> ps ) <-> ( ( ph -> ps ) /\ ( ps -> ph ) ) ) <-> ( ( ( ph <-> ps ) -> ( ( ph -> ps ) /\ ( ps -> ph ) ) ) /\ ( ( ( ph -> ps ) /\ ( ps -> ph ) ) -> ( ph <-> ps ) ) ) )
3	1 2	mpbi	|- ( ( ( ph <-> ps ) -> ( ( ph -> ps ) /\ ( ps -> ph ) ) ) /\ ( ( ( ph -> ps ) /\ ( ps -> ph ) ) -> ( ph <-> ps ) ) )