Theorem dfbi1 184

Description: Relate the biconditional connective to primitive connectives. See dfbi1gb 185 for an unusual version proved directly from axioms. (Contributed by NM, 5-Aug-1993.)

Assertion

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

Proof of Theorem dfbi1

Step	Hyp	Ref	Expression
1		df-bi 177	. . 3 |- -. (((ph <-> ps) -> -. ((ph -> ps) -> -. (ps -> ph))) -> -. (-. ((ph -> ps) -> -. (ps -> ph)) -> (ph <-> ps)))
2		simplim 143	. . 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		bi3 179	. . 3 |- ((ph -> ps) -> ((ps -> ph) -> (ph <-> ps)))
5	4	impi 140	. 2 |- (-. ((ph -> ps) -> -. (ps -> ph)) -> (ph <-> ps))
6	3, 5	impbii 180	1 |- ((ph <-> ps) <-> -. ((ph -> ps) -> -. (ps -> ph)))

This proof depends on syntax axioms:  -. wn 3   -> wi 4   <-> wb 176

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8

This proof depends on definitions:  df-bi 177

This theorem is used by:  bi2  189  dfbi2  609  tbw-bijust  1463  rb-bijust  1514  nfbidOLD  1833