Theorem an6 1332

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		df-3an 982	. . . 4 |- ( ( ph /\ ps /\ ch ) <-> ( ( ph /\ ps ) /\ ch ) )
2		df-3an 982	. . . 4 |- ( ( th /\ ta /\ et ) <-> ( ( th /\ ta ) /\ et ) )
3	1, 2	anbi12i 460	. . 3 |- ( ( ( ph /\ ps /\ ch ) /\ ( th /\ ta /\ et ) ) <-> ( ( ( ph /\ ps ) /\ ch ) /\ ( ( th /\ ta ) /\ et ) ) )
4		an4 586	. . 3 |- ( ( ( ( ph /\ ps ) /\ ch ) /\ ( ( th /\ ta ) /\ et ) ) <-> ( ( ( ph /\ ps ) /\ ( th /\ ta ) ) /\ ( ch /\ et ) ) )
5		an4 586	. . . 4 |- ( ( ( ph /\ ps ) /\ ( th /\ ta ) ) <-> ( ( ph /\ th ) /\ ( ps /\ ta ) ) )
6	5	anbi1i 458	. . 3 |- ( ( ( ( ph /\ ps ) /\ ( th /\ ta ) ) /\ ( ch /\ et ) ) <-> ( ( ( ph /\ th ) /\ ( ps /\ ta ) ) /\ ( ch /\ et ) ) )
7	3, 4, 6	3bitri 206	. 2 |- ( ( ( ph /\ ps /\ ch ) /\ ( th /\ ta /\ et ) ) <-> ( ( ( ph /\ th ) /\ ( ps /\ ta ) ) /\ ( ch /\ et ) ) )
8		df-3an 982	. 2 |- ( ( ( ph /\ th ) /\ ( ps /\ ta ) /\ ( ch /\ et ) ) <-> ( ( ( ph /\ th ) /\ ( ps /\ ta ) ) /\ ( ch /\ et ) ) )
9	7, 8	bitr4i 187	1 |- ( ( ( ph /\ ps /\ ch ) /\ ( th /\ ta /\ et ) ) <-> ( ( ph /\ th ) /\ ( ps /\ ta ) /\ ( ch /\ et ) ) )

Syntax hints:   /\ wa 104   <-> wb 105   /\ w3a 980

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108

This theorem depends on definitions:  df-bi 117  df-3an 982

This theorem is referenced by:  3an6  1333  elfzuzb  10111