Theorem pm5.32 617

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 286	. . . 4 |- ((ps <-> ch) <-> (-. ps <-> -. ch))
2	1	imbi2i 303	. . 3 |- ((ph -> (ps <-> ch)) <-> (ph -> (-. ps <-> -. ch)))
3		pm5.74 235	. . 3 |- ((ph -> (-. ps <-> -. ch)) <-> ((ph -> -. ps) <-> (ph -> -. ch)))
4		notbi 286	. . 3 |- (((ph -> -. ps) <-> (ph -> -. ch)) <-> (-. (ph -> -. ps) <-> -. (ph -> -. ch)))
5	2, 3, 4	3bitri 262	. 2 |- ((ph -> (ps <-> ch)) <-> (-. (ph -> -. ps) <-> -. (ph -> -. ch)))
6		df-an 360	. . 3 |- ((ph /\ ps) <-> -. (ph -> -. ps))
7		df-an 360	. . 3 |- ((ph /\ ch) <-> -. (ph -> -. ch))
8	6, 7	bibi12i 306	. 2 |- (((ph /\ ps) <-> (ph /\ ch)) <-> (-. (ph -> -. ps) <-> -. (ph -> -. ch)))
9	5, 8	bitr4i 243	1 |- ((ph -> (ps <-> ch)) <-> ((ph /\ ps) <-> (ph /\ ch)))

Syntax hints:  -. wn 3   -> wi 4   <-> wb 176   /\ wa 358

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  df-an 360

This theorem is referenced by:  pm5.32i  618  pm5.32d  620  xordi  865  cbval2  2004  cbvex2  2005  rabbi  2790