Theorem pm5.74 582

Description: Distribution of implication over biconditional. Theorem *5.74 of [WhiteheadRussell] p. 126.

Assertion

Ref	Expression
pm5.74	|- ((ph -> (ps <-> ch)) <-> ((ph -> ps) <-> (ph -> ch)))

Proof of Theorem pm5.74

Step	Hyp	Ref	Expression
1		bi1 148	. . . 4 |- ((ps <-> ch) -> (ps -> ch))
2	1	imim3i 19	. . 3 |- ((ph -> (ps <-> ch)) -> ((ph -> ps) -> (ph -> ch)))
3		bi2 149	. . . 4 |- ((ps <-> ch) -> (ch -> ps))
4	3	imim3i 19	. . 3 |- ((ph -> (ps <-> ch)) -> ((ph -> ch) -> (ph -> ps)))
5	2, 4	impbid 515	. 2 |- ((ph -> (ps <-> ch)) -> ((ph -> ps) <-> (ph -> ch)))
6		bi1 148	. . . . 5 |- (((ph -> ps) <-> (ph -> ch)) -> ((ph -> ps) -> (ph -> ch)))
7	6	pm2.86d 71	. . . 4 |- (((ph -> ps) <-> (ph -> ch)) -> (ph -> (ps -> ch)))
8		bi2 149	. . . . 5 |- (((ph -> ps) <-> (ph -> ch)) -> ((ph -> ch) -> (ph -> ps)))
9	8	pm2.86d 71	. . . 4 |- (((ph -> ps) <-> (ph -> ch)) -> (ph -> (ch -> ps)))
10	7, 9	anim12d 557	. . 3 |- (((ph -> ps) <-> (ph -> ch)) -> ((ph /\ ph) -> ((ps -> ch) /\ (ch -> ps))))
11		pm4.24 433	. . 3 |- (ph <-> (ph /\ ph))
12		bi 514	. . 3 |- ((ps <-> ch) <-> ((ps -> ch) /\ (ch -> ps)))
13	10, 11, 12	3imtr4g 552	. 2 |- (((ph -> ps) <-> (ph -> ch)) -> (ph -> (ps <-> ch)))
14	5, 13	impbi 157	1 |- ((ph -> (ps <-> ch)) <-> ((ph -> ps) <-> (ph -> ch)))

Syntax hints:   -> wi 3   <-> wb 146   /\ wa 223

This theorem is referenced by:  pm5.74i 583  pm5.74d 584  pm5.74ri 586  pm5.74rd 587  pm5.32 643

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-an 225

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