Theorem pm5.32 488

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		pm4.11 400	. . . 4 |- ((ps <-> ch) <-> (-. ps <-> -. ch))
2	1	imbi2i 160	. . 3 |- ((ph -> (ps <-> ch)) <-> (ph -> (-. ps <-> -. ch)))
3		pm5.74 442	. . 3 |- ((ph -> (-. ps <-> -. ch)) <-> ((ph -> -. ps) <-> (ph -> -. ch)))
4		pm4.11 400	. . 3 |- (((ph -> -. ps) <-> (ph -> -. ch)) <-> (-. (ph -> -. ps) <-> -. (ph -> -. ch)))
5	2, 3, 4	3bitr 155	. 2 |- ((ph -> (ps <-> ch)) <-> (-. (ph -> -. ps) <-> -. (ph -> -. ch)))
6		df-an 198	. . 3 |- ((ph /\ ps) <-> -. (ph -> -. ps))
7		df-an 198	. . 3 |- ((ph /\ ch) <-> -. (ph -> -. ch))
8	6, 7	bibi12i 462	. 2 |- (((ph /\ ps) <-> (ph /\ ch)) <-> (-. (ph -> -. ps) <-> -. (ph -> -. ch)))
9	5, 8	bitr4 154	1 |- ((ph -> (ps <-> ch)) <-> ((ph /\ ps) <-> (ph /\ ch)))

Syntax hints:  -. wn 1   -> wi 2   <-> wb 127   /\ wa 196

This theorem is referenced by:  pm5.32i 489  pm5.32d 491  cbval2 974  cbvex2 975

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

This theorem depends on definitions:  df-bi 128  df-an 198

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