Theorem pm5.74 235

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 178	. . . 4 |- ((ps <-> ch) -> (ps -> ch))
2	1	imim3i 55	. . 3 |- ((ph -> (ps <-> ch)) -> ((ph -> ps) -> (ph -> ch)))
3		bi2 189	. . . 4 |- ((ps <-> ch) -> (ch -> ps))
4	3	imim3i 55	. . 3 |- ((ph -> (ps <-> ch)) -> ((ph -> ch) -> (ph -> ps)))
5	2, 4	impbid 183	. 2 |- ((ph -> (ps <-> ch)) -> ((ph -> ps) <-> (ph -> ch)))
6		bi1 178	. . . 4 |- (((ph -> ps) <-> (ph -> ch)) -> ((ph -> ps) -> (ph -> ch)))
7	6	pm2.86d 93	. . 3 |- (((ph -> ps) <-> (ph -> ch)) -> (ph -> (ps -> ch)))
8		bi2 189	. . . 4 |- (((ph -> ps) <-> (ph -> ch)) -> ((ph -> ch) -> (ph -> ps)))
9	8	pm2.86d 93	. . 3 |- (((ph -> ps) <-> (ph -> ch)) -> (ph -> (ch -> ps)))
10	7, 9	impbidd 181	. 2 |- (((ph -> ps) <-> (ph -> ch)) -> (ph -> (ps <-> ch)))
11	5, 10	impbii 180	1 |- ((ph -> (ps <-> ch)) <-> ((ph -> ps) <-> (ph -> ch)))

Syntax hints:   -> wi 4   <-> wb 176

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

This theorem is referenced by:  pm5.74i  236  pm5.74ri  237  pm5.74d  238  pm5.74rd  239  bibi2d  309  pm5.32  617  orbidi  898