Theorem pm5.32 449

Description: Distribution of implication over biconditional. Theorem *5.32 of [WhiteheadRussell] p. 125. (Contributed by NM, 1-Aug-1994.) (Revised by NM, 31-Jan-2015.)

Assertion

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

Proof of Theorem pm5.32

Step	Hyp	Ref	Expression
1		id 19	. . 3 |- ( ( ph -> ( ps <-> ch ) ) -> ( ph -> ( ps <-> ch ) ) )
2	1	pm5.32d 446	. 2 |- ( ( ph -> ( ps <-> ch ) ) -> ( ( ph /\ ps ) <-> ( ph /\ ch ) ) )
3		ibar 299	. . . 4 |- ( ph -> ( ps <-> ( ph /\ ps ) ) )
4		ibar 299	. . . 4 |- ( ph -> ( ch <-> ( ph /\ ch ) ) )
5	3, 4	bibi12d 234	. . 3 |- ( ph -> ( ( ps <-> ch ) <-> ( ( ph /\ ps ) <-> ( ph /\ ch ) ) ) )
6	5	biimprcd 159	. 2 |- ( ( ( ph /\ ps ) <-> ( ph /\ ch ) ) -> ( ph -> ( ps <-> ch ) ) )
7	2, 6	impbii 125	1 |- ( ( ph -> ( ps <-> ch ) ) <-> ( ( ph /\ ps ) <-> ( ph /\ ch ) ) )

Syntax hints:   -> wi 4   /\ wa 103   <-> 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:  pm5.32i  450  biadani  602  xordidc  1389  cbvex2  1910  rabbi  2643  rabxfrd  4447  asymref  4989  rexrnmpt  5628  mpo2eqb  5951