Theorem an6 1408

Description: Rearrangement of 6 conjuncts. (Contributed by NM, 13-Mar-1995.)

Assertion

Ref	Expression
an6	|- ( ( ( ph /\ ps /\ ch ) /\ ( th /\ ta /\ et ) ) <-> ( ( ph /\ th ) /\ ( ps /\ ta ) /\ ( ch /\ et ) ) )

Proof of Theorem an6

Step	Hyp	Ref	Expression
1		an4 865	. . 3 |- ( ( ( ( ph /\ ps ) /\ ch ) /\ ( ( th /\ ta ) /\ et ) ) <-> ( ( ( ph /\ ps ) /\ ( th /\ ta ) ) /\ ( ch /\ et ) ) )
2		an4 865	. . . 4 |- ( ( ( ph /\ ps ) /\ ( th /\ ta ) ) <-> ( ( ph /\ th ) /\ ( ps /\ ta ) ) )
3	2	anbi1i 731	. . 3 |- ( ( ( ( ph /\ ps ) /\ ( th /\ ta ) ) /\ ( ch /\ et ) ) <-> ( ( ( ph /\ th ) /\ ( ps /\ ta ) ) /\ ( ch /\ et ) ) )
4	1, 3	bitri 264	. 2 |- ( ( ( ( ph /\ ps ) /\ ch ) /\ ( ( th /\ ta ) /\ et ) ) <-> ( ( ( ph /\ th ) /\ ( ps /\ ta ) ) /\ ( ch /\ et ) ) )
5		df-3an 1039	. . 3 |- ( ( ph /\ ps /\ ch ) <-> ( ( ph /\ ps ) /\ ch ) )
6		df-3an 1039	. . 3 |- ( ( th /\ ta /\ et ) <-> ( ( th /\ ta ) /\ et ) )
7	5, 6	anbi12i 733	. 2 |- ( ( ( ph /\ ps /\ ch ) /\ ( th /\ ta /\ et ) ) <-> ( ( ( ph /\ ps ) /\ ch ) /\ ( ( th /\ ta ) /\ et ) ) )
8		df-3an 1039	. 2 |- ( ( ( ph /\ th ) /\ ( ps /\ ta ) /\ ( ch /\ et ) ) <-> ( ( ( ph /\ th ) /\ ( ps /\ ta ) ) /\ ( ch /\ et ) ) )
9	4, 7, 8	3bitr4i 292	1 |- ( ( ( ph /\ ps /\ ch ) /\ ( th /\ ta /\ et ) ) <-> ( ( ph /\ th ) /\ ( ps /\ ta ) /\ ( ch /\ et ) ) )

Syntax hints:   <-> wb 196   /\ wa 384   /\ w3a 1037

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  df-3an 1039

This theorem is referenced by:  3an6  1409  elfzuzb  12336  fzadd2  12376  ptbasin  21380  iimulcl  22736  nb3grpr  26284  nb3grpr2  26285  txpconn  31214  paddasslem9  35114  paddasslem10  35115  gboge9  41652