Theorem pm5.32 668

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

Assertion

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

Proof of Theorem pm5.32

Step	Hyp	Ref	Expression
1		notbi 309	. . . 4 |- ( ( ps <-> ch ) <-> ( -. ps <-> -. ch ) )
2	1	imbi2i 326	. . 3 |- ( ( ph -> ( ps <-> ch ) ) <-> ( ph -> ( -. ps <-> -. ch ) ) )
3		pm5.74 259	. . 3 |- ( ( ph -> ( -. ps <-> -. ch ) ) <-> ( ( ph -> -. ps ) <-> ( ph -> -. ch ) ) )
4		notbi 309	. . 3 |- ( ( ( ph -> -. ps ) <-> ( ph -> -. ch ) ) <-> ( -. ( ph -> -. ps ) <-> -. ( ph -> -. ch ) ) )
5	2, 3, 4	3bitri 286	. 2 |- ( ( ph -> ( ps <-> ch ) ) <-> ( -. ( ph -> -. ps ) <-> -. ( ph -> -. ch ) ) )
6		df-an 386	. . 3 |- ( ( ph /\ ps ) <-> -. ( ph -> -. ps ) )
7		df-an 386	. . 3 |- ( ( ph /\ ch ) <-> -. ( ph -> -. ch ) )
8	6, 7	bibi12i 329	. 2 |- ( ( ( ph /\ ps ) <-> ( ph /\ ch ) ) <-> ( -. ( ph -> -. ps ) <-> -. ( ph -> -. ch ) ) )
9	5, 8	bitr4i 267	1 |- ( ( ph -> ( ps <-> ch ) ) <-> ( ( ph /\ ps ) <-> ( ph /\ ch ) ) )

Syntax hints:   -. wn 3   -> wi 4   <-> wb 196   /\ wa 384

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 197  df-an 386

This theorem is referenced by:  pm5.32i  669  pm5.32d  671  xordi  937  rabbi  3120  rabxfrd  4889  asymref  5512  mpt22eqb  6769  cfilucfil4  23118  wl-ax11-lem8  33369  relexp0eq  37993  2sb5nd  38776  2sb5ndVD  39146  2sb5ndALT  39168