Theorem pm5.74 442

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 130	. . . . 5 |- ((ps <-> ch) -> (ps -> ch))
2	1	syl3 18	. . . 4 |- ((ph -> (ps <-> ch)) -> (ph -> (ps -> ch)))
3	2	a2d 15	. . 3 |- ((ph -> (ps <-> ch)) -> ((ph -> ps) -> (ph -> ch)))
4		bi2 131	. . . . 5 |- ((ps <-> ch) -> (ch -> ps))
5	4	syl3 18	. . . 4 |- ((ph -> (ps <-> ch)) -> (ph -> (ch -> ps)))
6	5	a2d 15	. . 3 |- ((ph -> (ps <-> ch)) -> ((ph -> ch) -> (ph -> ps)))
7	3, 6	impbid 397	. 2 |- ((ph -> (ps <-> ch)) -> ((ph -> ps) <-> (ph -> ch)))
8		bi1 130	. . . . 5 |- (((ph -> ps) <-> (ph -> ch)) -> ((ph -> ps) -> (ph -> ch)))
9	8	pm2.86d 65	. . . 4 |- (((ph -> ps) <-> (ph -> ch)) -> (ph -> (ps -> ch)))
10		bi2 131	. . . . 5 |- (((ph -> ps) <-> (ph -> ch)) -> ((ph -> ch) -> (ph -> ps)))
11	10	pm2.86d 65	. . . 4 |- (((ph -> ps) <-> (ph -> ch)) -> (ph -> (ch -> ps)))
12	9, 11	anim12d 431	. . 3 |- (((ph -> ps) <-> (ph -> ch)) -> ((ph /\ ph) -> ((ps -> ch) /\ (ch -> ps))))
13		anidm 331	. . . 4 |- ((ph /\ ph) <-> ph)
14	13	bicomi 150	. . 3 |- (ph <-> (ph /\ ph))
15		bi 396	. . 3 |- ((ps <-> ch) <-> ((ps -> ch) /\ (ch -> ps)))
16	12, 14, 15	3imtr4g 426	. 2 |- (((ph -> ps) <-> (ph -> ch)) -> (ph -> (ps <-> ch)))
17	7, 16	impbi 139	1 |- ((ph -> (ps <-> ch)) <-> ((ph -> ps) <-> (ph -> ch)))

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

This theorem is referenced by:  pm5.74i 443  pm5.74d 444  pm5.74ri 445  pm5.74rd 446  pm5.32 488

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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