Theorem bi3 179

Description: Property of the biconditional connective. (Contributed by NM, 11-May-1999.)

Assertion

Ref	Expression
bi3	|- ((ph -> ps) -> ((ps -> ph) -> (ph <-> ps)))

Proof of Theorem bi3

Step	Hyp	Ref	Expression
1		df-bi 177	. . 3 |- -. (((ph <-> ps) -> -. ((ph -> ps) -> -. (ps -> ph))) -> -. (-. ((ph -> ps) -> -. (ps -> ph)) -> (ph <-> ps)))
2		simprim 142	. . 3 |- (-. (((ph <-> ps) -> -. ((ph -> ps) -> -. (ps -> ph))) -> -. (-. ((ph -> ps) -> -. (ps -> ph)) -> (ph <-> ps))) -> (-. ((ph -> ps) -> -. (ps -> ph)) -> (ph <-> ps)))
3	1, 2	ax-mp 5	. 2 |- (-. ((ph -> ps) -> -. (ps -> ph)) -> (ph <-> ps))
4	3	expi 141	1 |- ((ph -> ps) -> ((ps -> ph) -> (ph <-> ps)))

Syntax hints:  -. wn 3   -> wi 4   <-> wb 176

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 177

This theorem is referenced by:  impbii  180  impbidd  181  dfbi1  184  bisym  281