Theorem pm5.32

Description: Distribution of implication over biconditional. Theorem *5.32 of WhiteheadRussell p. 125. (Contributed by NM, 1-Aug-1994)

	Ref	Expression
Assertion	pm5.32	|- ( ( ph -> ( ps <-> ch ) ) <-> ( ( ph /\ ps ) <-> ( ph /\ ch ) ) )

Proof

Step	Hyp	Ref	Expression
1		notbi	|- ( ( ps <-> ch ) <-> ( -. ps <-> -. ch ) )
2	1	imbi2i	|- ( ( ph -> ( ps <-> ch ) ) <-> ( ph -> ( -. ps <-> -. ch ) ) )
3		pm5.74	|- ( ( ph -> ( -. ps <-> -. ch ) ) <-> ( ( ph -> -. ps ) <-> ( ph -> -. ch ) ) )
4		notbi	|- ( ( ( ph -> -. ps ) <-> ( ph -> -. ch ) ) <-> ( -. ( ph -> -. ps ) <-> -. ( ph -> -. ch ) ) )
5	2 3 4	3bitri	|- ( ( ph -> ( ps <-> ch ) ) <-> ( -. ( ph -> -. ps ) <-> -. ( ph -> -. ch ) ) )
6		df-an	|- ( ( ph /\ ps ) <-> -. ( ph -> -. ps ) )
7		df-an	|- ( ( ph /\ ch ) <-> -. ( ph -> -. ch ) )
8	6 7	bibi12i	|- ( ( ( ph /\ ps ) <-> ( ph /\ ch ) ) <-> ( -. ( ph -> -. ps ) <-> -. ( ph -> -. ch ) ) )
9	5 8	bitr4i	|- ( ( ph -> ( ps <-> ch ) ) <-> ( ( ph /\ ps ) <-> ( ph /\ ch ) ) )