Theorem pm4.78 354

Description: Theorem *4.78 of [WhiteheadRussell] p. 121.

Assertion

Ref	Expression
pm4.78	|- (((ph -> ps) \/ (ph -> ch)) <-> (ph -> (ps \/ ch)))

Proof of Theorem pm4.78

Step	Hyp	Ref	Expression
1		impexp 347	. . 3 |- (((ph /\ -. ps) -> (ph -> ch)) <-> (ph -> (-. ps -> (ph -> ch))))
2		annim 238	. . . 4 |- ((ph /\ -. ps) <-> -. (ph -> ps))
3	2	imbi1i 186	. . 3 |- (((ph /\ -. ps) -> (ph -> ch)) <-> (-. (ph -> ps) -> (ph -> ch)))
4		bi2.04 160	. . . . 5 |- ((-. ps -> (ph -> ch)) <-> (ph -> (-. ps -> ch)))
5	4	imbi2i 185	. . . 4 |- ((ph -> (-. ps -> (ph -> ch))) <-> (ph -> (ph -> (-. ps -> ch))))
6		pm5.4 167	. . . 4 |- ((ph -> (ph -> (-. ps -> ch))) <-> (ph -> (-. ps -> ch)))
7	5, 6	bitr 173	. . 3 |- ((ph -> (-. ps -> (ph -> ch))) <-> (ph -> (-. ps -> ch)))
8	1, 3, 7	3bitr3 181	. 2 |- ((-. (ph -> ps) -> (ph -> ch)) <-> (ph -> (-. ps -> ch)))
9		df-or 224	. 2 |- (((ph -> ps) \/ (ph -> ch)) <-> (-. (ph -> ps) -> (ph -> ch)))
10		df-or 224	. . 3 |- ((ps \/ ch) <-> (-. ps -> ch))
11	10	imbi2i 185	. 2 |- ((ph -> (ps \/ ch)) <-> (ph -> (-. ps -> ch)))
12	8, 9, 11	3bitr4 183	1 |- (((ph -> ps) \/ (ph -> ch)) <-> (ph -> (ps \/ ch)))

Syntax hints:  -. wn 2   -> wi 3   <-> wb 146   \/ wo 222   /\ wa 223

This theorem is referenced by:  pm4.79 355

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 147  df-or 224  df-an 225

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