Theorem pm5.75 903

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.)

Assertion

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

Proof of Theorem pm5.75

Step	Hyp	Ref	Expression
1		anbi1 687	. . 3 |- ((ph <-> (ps \/ ch)) -> ((ph /\ -. ps) <-> ((ps \/ ch) /\ -. ps)))
2		orcom 376	. . . . 5 |- ((ps \/ ch) <-> (ch \/ ps))
3	2	anbi1i 676	. . . 4 |- (((ps \/ ch) /\ -. ps) <-> ((ch \/ ps) /\ -. ps))
4		pm5.61 693	. . . 4 |- (((ch \/ ps) /\ -. ps) <-> (ch /\ -. ps))
5	3, 4	bitri 240	. . 3 |- (((ps \/ ch) /\ -. ps) <-> (ch /\ -. ps))
6	1, 5	syl6bb 252	. 2 |- ((ph <-> (ps \/ ch)) -> ((ph /\ -. ps) <-> (ch /\ -. ps)))
7		pm4.71 611	. . . 4 |- ((ch -> -. ps) <-> (ch <-> (ch /\ -. ps)))
8	7	biimpi 186	. . 3 |- ((ch -> -. ps) -> (ch <-> (ch /\ -. ps)))
9	8	bicomd 192	. 2 |- ((ch -> -. ps) -> ((ch /\ -. ps) <-> ch))
10	6, 9	sylan9bbr 681	1 |- (((ch -> -. ps) /\ (ph <-> (ps \/ ch))) -> ((ph /\ -. ps) <-> ch))

Syntax hints:  -. wn 3   -> wi 4   <-> wb 176   \/ wo 357   /\ wa 358

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  df-or 359  df-an 360

This theorem is referenced by: (None)