Theorem pm5.75 978

Description: Theorem *5.75 of [WhiteheadRussell] p. 126. (Contributed by NM, 3-Jan-2005.) (Proof shortened by Andrew Salmon, 7-May-2011.) (Proof shortened by Wolf Lammen, 23-Dec-2012.) (Proof shortened by Kyle Wyonch, 12-Feb-2021.)

Assertion

Ref	Expression
pm5.75	|- ( ( ( ch -> -. ps ) /\ ( ph <-> ( ps \/ ch ) ) ) -> ( ( ph /\ -. ps ) <-> ch ) )

Proof of Theorem pm5.75

Step	Hyp	Ref	Expression
1		anbi1 743	. . 3 |- ( ( ph <-> ( ps \/ ch ) ) -> ( ( ph /\ -. ps ) <-> ( ( ps \/ ch ) /\ -. ps ) ) )
2		biorf 420	. . . . 5 |- ( -. ps -> ( ch <-> ( ps \/ ch ) ) )
3	2	bicomd 213	. . . 4 |- ( -. ps -> ( ( ps \/ ch ) <-> ch ) )
4	3	pm5.32ri 670	. . 3 |- ( ( ( ps \/ ch ) /\ -. ps ) <-> ( ch /\ -. ps ) )
5	1, 4	syl6bb 276	. 2 |- ( ( ph <-> ( ps \/ ch ) ) -> ( ( ph /\ -. ps ) <-> ( ch /\ -. ps ) ) )
6		pm4.71 662	. . . 4 |- ( ( ch -> -. ps ) <-> ( ch <-> ( ch /\ -. ps ) ) )
7	6	biimpi 206	. . 3 |- ( ( ch -> -. ps ) -> ( ch <-> ( ch /\ -. ps ) ) )
8	7	bicomd 213	. 2 |- ( ( ch -> -. ps ) -> ( ( ch /\ -. ps ) <-> ch ) )
9	5, 8	sylan9bbr 737	1 |- ( ( ( ch -> -. ps ) /\ ( ph <-> ( ps \/ ch ) ) ) -> ( ( ph /\ -. ps ) <-> ch ) )

Syntax hints:   -. wn 3   -> wi 4   <-> wb 196   \/ wo 383   /\ wa 384

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  df-or 385  df-an 386

This theorem is referenced by: (None)