Theorem an6 1215

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 886	. . . 4 |- ((ph /\ ps /\ ch) <-> ((ph /\ ps) /\ ch))
2		df-3an 886	. . . 4 |- ((th /\ ta /\ et) <-> ((th /\ ta) /\ et))
3	1, 2	anbi12i 433	. . 3 |- (((ph /\ ps /\ ch) /\ (th /\ ta /\ et)) <-> (((ph /\ ps) /\ ch) /\ ((th /\ ta) /\ et)))
4		an4 520	. . 3 |- ((((ph /\ ps) /\ ch) /\ ((th /\ ta) /\ et)) <-> (((ph /\ ps) /\ (th /\ ta)) /\ (ch /\ et)))
5		an4 520	. . . 4 |- (((ph /\ ps) /\ (th /\ ta)) <-> ((ph /\ th) /\ (ps /\ ta)))
6	5	anbi1i 431	. . 3 |- ((((ph /\ ps) /\ (th /\ ta)) /\ (ch /\ et)) <-> (((ph /\ th) /\ (ps /\ ta)) /\ (ch /\ et)))
7	3, 4, 6	3bitri 195	. 2 |- (((ph /\ ps /\ ch) /\ (th /\ ta /\ et)) <-> (((ph /\ th) /\ (ps /\ ta)) /\ (ch /\ et)))
8		df-3an 886	. 2 |- (((ph /\ th) /\ (ps /\ ta) /\ (ch /\ et)) <-> (((ph /\ th) /\ (ps /\ ta)) /\ (ch /\ et)))
9	7, 8	bitr4i 176	1 |- (((ph /\ ps /\ ch) /\ (th /\ ta /\ et)) <-> ((ph /\ th) /\ (ps /\ ta) /\ (ch /\ et)))

Syntax hints:   /\ wa 97   <-> wb 98   /\ w3a 884

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 99  ax-ia2 100  ax-ia3 101

This theorem depends on definitions:  df-bi 110  df-3an 886

This theorem is referenced by:  3an6  1216  elfzuzb  8654