Theorem dfbi1

Description: Relate the biconditional connective to primitive connectives. See dfbi1ALT for an unusual version proved directly from axioms. (Contributed by NM, 29-Dec-1992)

	Ref	Expression
Assertion	dfbi1	|- ( ( ph <-> ps ) <-> -. ( ( ph -> ps ) -> -. ( ps -> ph ) ) )

Proof

Step	Hyp	Ref	Expression
1		df-bi	|- -. ( ( ( ph <-> ps ) -> -. ( ( ph -> ps ) -> -. ( ps -> ph ) ) ) -> -. ( -. ( ( ph -> ps ) -> -. ( ps -> ph ) ) -> ( ph <-> ps ) ) )
2		impbi	|- ( ( ( ph <-> ps ) -> -. ( ( ph -> ps ) -> -. ( ps -> ph ) ) ) -> ( ( -. ( ( ph -> ps ) -> -. ( ps -> ph ) ) -> ( ph <-> ps ) ) -> ( ( ph <-> ps ) <-> -. ( ( ph -> ps ) -> -. ( ps -> ph ) ) ) ) )
3	2	con3rr3	|- ( -. ( ( ph <-> ps ) <-> -. ( ( ph -> ps ) -> -. ( ps -> ph ) ) ) -> ( ( ( ph <-> ps ) -> -. ( ( ph -> ps ) -> -. ( ps -> ph ) ) ) -> -. ( -. ( ( ph -> ps ) -> -. ( ps -> ph ) ) -> ( ph <-> ps ) ) ) )
4	1 3	mt3	|- ( ( ph <-> ps ) <-> -. ( ( ph -> ps ) -> -. ( ps -> ph ) ) )