Theorem pm5.74 178

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.)

Assertion

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

Proof of Theorem pm5.74

Step	Hyp	Ref	Expression
1		bi1 117	. . . 4 |- ( ( ps <-> ch ) -> ( ps -> ch ) )
2	1	imim3i 61	. . 3 |- ( ( ph -> ( ps <-> ch ) ) -> ( ( ph -> ps ) -> ( ph -> ch ) ) )
3		bi2 129	. . . 4 |- ( ( ps <-> ch ) -> ( ch -> ps ) )
4	3	imim3i 61	. . 3 |- ( ( ph -> ( ps <-> ch ) ) -> ( ( ph -> ch ) -> ( ph -> ps ) ) )
5	2, 4	impbid 128	. 2 |- ( ( ph -> ( ps <-> ch ) ) -> ( ( ph -> ps ) <-> ( ph -> ch ) ) )
6		bi1 117	. . . 4 |- ( ( ( ph -> ps ) <-> ( ph -> ch ) ) -> ( ( ph -> ps ) -> ( ph -> ch ) ) )
7	6	pm2.86d 99	. . 3 |- ( ( ( ph -> ps ) <-> ( ph -> ch ) ) -> ( ph -> ( ps -> ch ) ) )
8		bi2 129	. . . 4 |- ( ( ( ph -> ps ) <-> ( ph -> ch ) ) -> ( ( ph -> ch ) -> ( ph -> ps ) ) )
9	8	pm2.86d 99	. . 3 |- ( ( ( ph -> ps ) <-> ( ph -> ch ) ) -> ( ph -> ( ch -> ps ) ) )
10	7, 9	impbidd 126	. 2 |- ( ( ( ph -> ps ) <-> ( ph -> ch ) ) -> ( ph -> ( ps <-> ch ) ) )
11	5, 10	impbii 125	1 |- ( ( ph -> ( ps <-> ch ) ) <-> ( ( ph -> ps ) <-> ( ph -> ch ) ) )

Syntax hints:   -> wi 4   <-> wb 104

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107

This theorem depends on definitions:  df-bi 116

This theorem is referenced by:  pm5.74i  179  pm5.74ri  180  pm5.74d  181  pm5.74rd  182  bibi2d  231