Theorem pm5.74

Description: Distribution of implication over biconditional. Theorem *5.74 of WhiteheadRussell p. 126. (Contributed by NM, 1-Aug-1994) (Proof shortened by Wolf Lammen, 11-Apr-2013)

	Ref	Expression
Assertion	pm5.74	|- ( ( ph -> ( ps <-> ch ) ) <-> ( ( ph -> ps ) <-> ( ph -> ch ) ) )

Proof

Step	Hyp	Ref	Expression
1		biimp	|- ( ( ps <-> ch ) -> ( ps -> ch ) )
2	1	imim3i	|- ( ( ph -> ( ps <-> ch ) ) -> ( ( ph -> ps ) -> ( ph -> ch ) ) )
3		biimpr	|- ( ( ps <-> ch ) -> ( ch -> ps ) )
4	3	imim3i	|- ( ( ph -> ( ps <-> ch ) ) -> ( ( ph -> ch ) -> ( ph -> ps ) ) )
5	2 4	impbid	|- ( ( ph -> ( ps <-> ch ) ) -> ( ( ph -> ps ) <-> ( ph -> ch ) ) )
6		biimp	|- ( ( ( ph -> ps ) <-> ( ph -> ch ) ) -> ( ( ph -> ps ) -> ( ph -> ch ) ) )
7	6	pm2.86d	|- ( ( ( ph -> ps ) <-> ( ph -> ch ) ) -> ( ph -> ( ps -> ch ) ) )
8		biimpr	|- ( ( ( ph -> ps ) <-> ( ph -> ch ) ) -> ( ( ph -> ch ) -> ( ph -> ps ) ) )
9	8	pm2.86d	|- ( ( ( ph -> ps ) <-> ( ph -> ch ) ) -> ( ph -> ( ch -> ps ) ) )
10	7 9	impbidd	|- ( ( ( ph -> ps ) <-> ( ph -> ch ) ) -> ( ph -> ( ps <-> ch ) ) )
11	5 10	impbii	|- ( ( ph -> ( ps <-> ch ) ) <-> ( ( ph -> ps ) <-> ( ph -> ch ) ) )