Theorem pm5.74 643

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 165	. . . 4 |- ((ps <-> ch) -> (ps -> ch))
2	1	imim3i 23	. . 3 |- ((ph -> (ps <-> ch)) -> ((ph -> ps) -> (ph -> ch)))
3		bi2 166	. . . 4 |- ((ps <-> ch) -> (ch -> ps))
4	3	imim3i 23	. . 3 |- ((ph -> (ps <-> ch)) -> ((ph -> ch) -> (ph -> ps)))
5	2, 4	impbid 574	. 2 |- ((ph -> (ps <-> ch)) -> ((ph -> ps) <-> (ph -> ch)))
6		bi1 165	. . . . 5 |- (((ph -> ps) <-> (ph -> ch)) -> ((ph -> ps) -> (ph -> ch)))
7	6	pm2.86d 87	. . . 4 |- (((ph -> ps) <-> (ph -> ch)) -> (ph -> (ps -> ch)))
8		bi2 166	. . . . 5 |- (((ph -> ps) <-> (ph -> ch)) -> ((ph -> ch) -> (ph -> ps)))
9	8	pm2.86d 87	. . . 4 |- (((ph -> ps) <-> (ph -> ch)) -> (ph -> (ch -> ps)))
10	7, 9	anim12d 617	. . 3 |- (((ph -> ps) <-> (ph -> ch)) -> ((ph /\ ph) -> ((ps -> ch) /\ (ch -> ps))))
11		pm4.24 479	. . 3 |- (ph <-> (ph /\ ph))
12		dfbi2 572	. . 3 |- ((ps <-> ch) <-> ((ps -> ch) /\ (ch -> ps)))
13	10, 11, 12	3imtr4g 612	. 2 |- (((ph -> ps) <-> (ph -> ch)) -> (ph -> (ps <-> ch)))
14	5, 13	impbii 174	1 |- ((ph -> (ps <-> ch)) <-> ((ph -> ps) <-> (ph -> ch)))

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

This theorem is referenced by:  pm5.74i 644  pm5.74d 645  pm5.74ri 647  pm5.74rd 648  pm5.32 706

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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