Theorem an6 1261

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 797	. . 3 |- ((((ph /\ ps) /\ ch) /\ ((th /\ ta) /\ et)) <-> (((ph /\ ps) /\ (th /\ ta)) /\ (ch /\ et)))
2		an4 797	. . . 4 |- (((ph /\ ps) /\ (th /\ ta)) <-> ((ph /\ th) /\ (ps /\ ta)))
3	2	anbi1i 676	. . 3 |- ((((ph /\ ps) /\ (th /\ ta)) /\ (ch /\ et)) <-> (((ph /\ th) /\ (ps /\ ta)) /\ (ch /\ et)))
4	1, 3	bitri 240	. 2 |- ((((ph /\ ps) /\ ch) /\ ((th /\ ta) /\ et)) <-> (((ph /\ th) /\ (ps /\ ta)) /\ (ch /\ et)))
5		df-3an 936	. . 3 |- ((ph /\ ps /\ ch) <-> ((ph /\ ps) /\ ch))
6		df-3an 936	. . 3 |- ((th /\ ta /\ et) <-> ((th /\ ta) /\ et))
7	5, 6	anbi12i 678	. 2 |- (((ph /\ ps /\ ch) /\ (th /\ ta /\ et)) <-> (((ph /\ ps) /\ ch) /\ ((th /\ ta) /\ et)))
8		df-3an 936	. 2 |- (((ph /\ th) /\ (ps /\ ta) /\ (ch /\ et)) <-> (((ph /\ th) /\ (ps /\ ta)) /\ (ch /\ et)))
9	4, 7, 8	3bitr4i 268	1 |- (((ph /\ ps /\ ch) /\ (th /\ ta /\ et)) <-> ((ph /\ th) /\ (ps /\ ta) /\ (ch /\ et)))

Syntax hints:   <-> wb 176   /\ wa 358   /\ w3a 934

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 177  df-an 360  df-3an 936

This theorem is referenced by:  3an6  1262  fntxp  5805