Theorem prtlem70 34141

Description: Lemma for prter3 34167: a rearrangement of conjuncts. (Contributed by Rodolfo Medina, 20-Oct-2010.)

Assertion

Ref	Expression
prtlem70	|- ( ( ( ( ps /\ et ) /\ ( ( ph /\ th ) /\ ( ch /\ ta ) ) ) /\ ph ) <-> ( ( ph /\ ( ps /\ ( ch /\ ( th /\ ta ) ) ) ) /\ et ) )

Proof of Theorem prtlem70

Step	Hyp	Ref	Expression
1		anass 681	. . . 4 |- ( ( ( ( ph /\ ps ) /\ ( ph /\ th ) ) /\ ( ch /\ ta ) ) <-> ( ( ph /\ ps ) /\ ( ( ph /\ th ) /\ ( ch /\ ta ) ) ) )
2	1	anbi1i 731	. . 3 |- ( ( ( ( ( ph /\ ps ) /\ ( ph /\ th ) ) /\ ( ch /\ ta ) ) /\ et ) <-> ( ( ( ph /\ ps ) /\ ( ( ph /\ th ) /\ ( ch /\ ta ) ) ) /\ et ) )
3		anandi 871	. . . . 5 |- ( ( ph /\ ( ps /\ th ) ) <-> ( ( ph /\ ps ) /\ ( ph /\ th ) ) )
4	3	anbi1i 731	. . . 4 |- ( ( ( ph /\ ( ps /\ th ) ) /\ ( ch /\ ta ) ) <-> ( ( ( ph /\ ps ) /\ ( ph /\ th ) ) /\ ( ch /\ ta ) ) )
5	4	anbi1i 731	. . 3 |- ( ( ( ( ph /\ ( ps /\ th ) ) /\ ( ch /\ ta ) ) /\ et ) <-> ( ( ( ( ph /\ ps ) /\ ( ph /\ th ) ) /\ ( ch /\ ta ) ) /\ et ) )
6		anass 681	. . . . 5 |- ( ( ( ph /\ ( ps /\ et ) ) /\ ( ( ph /\ th ) /\ ( ch /\ ta ) ) ) <-> ( ph /\ ( ( ps /\ et ) /\ ( ( ph /\ th ) /\ ( ch /\ ta ) ) ) ) )
7		anass 681	. . . . . 6 |- ( ( ( ph /\ ps ) /\ et ) <-> ( ph /\ ( ps /\ et ) ) )
8	7	anbi1i 731	. . . . 5 |- ( ( ( ( ph /\ ps ) /\ et ) /\ ( ( ph /\ th ) /\ ( ch /\ ta ) ) ) <-> ( ( ph /\ ( ps /\ et ) ) /\ ( ( ph /\ th ) /\ ( ch /\ ta ) ) ) )
9		ancom 466	. . . . 5 |- ( ( ( ( ps /\ et ) /\ ( ( ph /\ th ) /\ ( ch /\ ta ) ) ) /\ ph ) <-> ( ph /\ ( ( ps /\ et ) /\ ( ( ph /\ th ) /\ ( ch /\ ta ) ) ) ) )
10	6, 8, 9	3bitr4ri 293	. . . 4 |- ( ( ( ( ps /\ et ) /\ ( ( ph /\ th ) /\ ( ch /\ ta ) ) ) /\ ph ) <-> ( ( ( ph /\ ps ) /\ et ) /\ ( ( ph /\ th ) /\ ( ch /\ ta ) ) ) )
11		ancom 466	. . . . 5 |- ( ( ( ph /\ ps ) /\ et ) <-> ( et /\ ( ph /\ ps ) ) )
12	11	anbi1i 731	. . . 4 |- ( ( ( ( ph /\ ps ) /\ et ) /\ ( ( ph /\ th ) /\ ( ch /\ ta ) ) ) <-> ( ( et /\ ( ph /\ ps ) ) /\ ( ( ph /\ th ) /\ ( ch /\ ta ) ) ) )
13		anass 681	. . . . 5 |- ( ( ( et /\ ( ph /\ ps ) ) /\ ( ( ph /\ th ) /\ ( ch /\ ta ) ) ) <-> ( et /\ ( ( ph /\ ps ) /\ ( ( ph /\ th ) /\ ( ch /\ ta ) ) ) ) )
14		ancom 466	. . . . 5 |- ( ( et /\ ( ( ph /\ ps ) /\ ( ( ph /\ th ) /\ ( ch /\ ta ) ) ) ) <-> ( ( ( ph /\ ps ) /\ ( ( ph /\ th ) /\ ( ch /\ ta ) ) ) /\ et ) )
15	13, 14	bitri 264	. . . 4 |- ( ( ( et /\ ( ph /\ ps ) ) /\ ( ( ph /\ th ) /\ ( ch /\ ta ) ) ) <-> ( ( ( ph /\ ps ) /\ ( ( ph /\ th ) /\ ( ch /\ ta ) ) ) /\ et ) )
16	10, 12, 15	3bitri 286	. . 3 |- ( ( ( ( ps /\ et ) /\ ( ( ph /\ th ) /\ ( ch /\ ta ) ) ) /\ ph ) <-> ( ( ( ph /\ ps ) /\ ( ( ph /\ th ) /\ ( ch /\ ta ) ) ) /\ et ) )
17	2, 5, 16	3bitr4ri 293	. 2 |- ( ( ( ( ps /\ et ) /\ ( ( ph /\ th ) /\ ( ch /\ ta ) ) ) /\ ph ) <-> ( ( ( ph /\ ( ps /\ th ) ) /\ ( ch /\ ta ) ) /\ et ) )
18		anass 681	. . 3 |- ( ( ( ph /\ ( ps /\ th ) ) /\ ( ch /\ ta ) ) <-> ( ph /\ ( ( ps /\ th ) /\ ( ch /\ ta ) ) ) )
19	18	anbi1i 731	. 2 |- ( ( ( ( ph /\ ( ps /\ th ) ) /\ ( ch /\ ta ) ) /\ et ) <-> ( ( ph /\ ( ( ps /\ th ) /\ ( ch /\ ta ) ) ) /\ et ) )
20		an4 865	. . . . 5 |- ( ( ( ps /\ th ) /\ ( ch /\ ta ) ) <-> ( ( ps /\ ch ) /\ ( th /\ ta ) ) )
21		anass 681	. . . . 5 |- ( ( ( ps /\ ch ) /\ ( th /\ ta ) ) <-> ( ps /\ ( ch /\ ( th /\ ta ) ) ) )
22	20, 21	bitri 264	. . . 4 |- ( ( ( ps /\ th ) /\ ( ch /\ ta ) ) <-> ( ps /\ ( ch /\ ( th /\ ta ) ) ) )
23	22	anbi2i 730	. . 3 |- ( ( ph /\ ( ( ps /\ th ) /\ ( ch /\ ta ) ) ) <-> ( ph /\ ( ps /\ ( ch /\ ( th /\ ta ) ) ) ) )
24	23	anbi1i 731	. 2 |- ( ( ( ph /\ ( ( ps /\ th ) /\ ( ch /\ ta ) ) ) /\ et ) <-> ( ( ph /\ ( ps /\ ( ch /\ ( th /\ ta ) ) ) ) /\ et ) )
25	17, 19, 24	3bitri 286	1 |- ( ( ( ( ps /\ et ) /\ ( ( ph /\ th ) /\ ( ch /\ ta ) ) ) /\ ph ) <-> ( ( ph /\ ( ps /\ ( ch /\ ( th /\ ta ) ) ) ) /\ et ) )

Syntax hints:   <-> wb 196   /\ wa 384

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8

This theorem depends on definitions:  df-bi 197  df-an 386

This theorem is referenced by: (None)