Theorem dfbi1 203

Description: Relate the biconditional connective to primitive connectives. See dfbi1ALT 204 for an unusual version proved directly from axioms. (Contributed by NM, 29-Dec-1992.)

Assertion

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

Proof of Theorem dfbi1

Step	Hyp	Ref	Expression
1		df-bi 197	. . 3 |- -. ( ( ( ph <-> ps ) -> -. ( ( ph -> ps ) -> -. ( ps -> ph ) ) ) -> -. ( -. ( ( ph -> ps ) -> -. ( ps -> ph ) ) -> ( ph <-> ps ) ) )
2		simplim 163	. . 3 |- ( -. ( ( ( ph <-> ps ) -> -. ( ( ph -> ps ) -> -. ( ps -> ph ) ) ) -> -. ( -. ( ( ph -> ps ) -> -. ( ps -> ph ) ) -> ( ph <-> ps ) ) ) -> ( ( ph <-> ps ) -> -. ( ( ph -> ps ) -> -. ( ps -> ph ) ) ) )
3	1, 2	ax-mp 5	. 2 |- ( ( ph <-> ps ) -> -. ( ( ph -> ps ) -> -. ( ps -> ph ) ) )
4		impbi 198	. . 3 |- ( ( ph -> ps ) -> ( ( ps -> ph ) -> ( ph <-> ps ) ) )
5	4	impi 160	. 2 |- ( -. ( ( ph -> ps ) -> -. ( ps -> ph ) ) -> ( ph <-> ps ) )
6	3, 5	impbii 199	1 |- ( ( ph <-> ps ) <-> -. ( ( ph -> ps ) -> -. ( ps -> ph ) ) )

Syntax hints:   -. wn 3   -> wi 4   <-> wb 196

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8

This theorem depends on definitions:  df-bi 197

This theorem is referenced by:  biimpr  210  dfbi2  660  tbw-bijust  1623  rb-bijust  1674  axrepprim  31579  axacprim  31584