Theorem pm5.32 706

Description: Distribution of implication over biconditional. Theorem *5.32 of [WhiteheadRussell] p. 125.

Assertion

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

Proof of Theorem pm5.32

Step	Hyp	Ref	Expression
1		notbi 581	. . . 4 |- ((ps <-> ch) <-> (-. ps <-> -. ch))
2	1	imbi2i 202	. . 3 |- ((ph -> (ps <-> ch)) <-> (ph -> (-. ps <-> -. ch)))
3		pm5.74 643	. . 3 |- ((ph -> (-. ps <-> -. ch)) <-> ((ph -> -. ps) <-> (ph -> -. ch)))
4		notbi 581	. . 3 |- (((ph -> -. ps) <-> (ph -> -. ch)) <-> (-. (ph -> -. ps) <-> -. (ph -> -. ch)))
5	2, 3, 4	3bitri 194	. 2 |- ((ph -> (ps <-> ch)) <-> (-. (ph -> -. ps) <-> -. (ph -> -. ch)))
6		df-an 242	. . 3 |- ((ph /\ ps) <-> -. (ph -> -. ps))
7		df-an 242	. . 3 |- ((ph /\ ch) <-> -. (ph -> -. ch))
8	6, 7	bibi12i 672	. 2 |- (((ph /\ ps) <-> (ph /\ ch)) <-> (-. (ph -> -. ps) <-> -. (ph -> -. ch)))
9	5, 8	bitr4i 193	1 |- ((ph -> (ps <-> ch)) <-> ((ph /\ ps) <-> (ph /\ ch)))

Syntax hints:  -. wn 2   -> wi 3   <-> wb 163   /\ wa 240

This theorem is referenced by:  pm5.32i 707  pm5.32d 709  xordi 738  cbval2 1698  cbvex2 1699  rabxfrd 3842  asymref 4308  asymrefOLD 4309  eluz 7595  metcn 9167  flimopn 10321  dvdslelem 13692

This theorem was proved from axioms:  ax-1 4  ax-2 5  ax-3 6  ax-mp 7

This theorem depends on definitions:  df-bi 164  df-an 242

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