Theorem pm5.75 869

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 439	. . 3 |- ( ( ph <-> ( ps \/ ch ) ) -> ( ( ph /\ -. ps ) <-> ( ( ps \/ ch ) /\ -. ps ) ) )
2		orcom 647	. . . . 5 |- ( ( ps \/ ch ) <-> ( ch \/ ps ) )
3	2	anbi1i 431	. . . 4 |- ( ( ( ps \/ ch ) /\ -. ps ) <-> ( ( ch \/ ps ) /\ -. ps ) )
4		pm5.61 708	. . . 4 |- ( ( ( ch \/ ps ) /\ -. ps ) <-> ( ch /\ -. ps ) )
5	3, 4	bitri 173	. . 3 |- ( ( ( ps \/ ch ) /\ -. ps ) <-> ( ch /\ -. ps ) )
6	1, 5	syl6bb 185	. 2 |- ( ( ph <-> ( ps \/ ch ) ) -> ( ( ph /\ -. ps ) <-> ( ch /\ -. ps ) ) )
7		pm4.71 369	. . . 4 |- ( ( ch -> -. ps ) <-> ( ch <-> ( ch /\ -. ps ) ) )
8	7	biimpi 113	. . 3 |- ( ( ch -> -. ps ) -> ( ch <-> ( ch /\ -. ps ) ) )
9	8	bicomd 129	. 2 |- ( ( ch -> -. ps ) -> ( ( ch /\ -. ps ) <-> ch ) )
10	6, 9	sylan9bbr 436	1 |- ( ( ( ch -> -. ps ) /\ ( ph <-> ( ps \/ ch ) ) ) -> ( ( ph /\ -. ps ) <-> ch ) )

Syntax hints:   -. wn 3   -> wi 4   /\ wa 97   <-> wb 98   \/ wo 629

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 99  ax-ia2 100  ax-ia3 101  ax-in2 545  ax-io 630

This theorem depends on definitions:  df-bi 110

This theorem is referenced by: (None)