Theorem an6 638

Description: Rearrangement of 6 conjuncts.

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 583	. . . 4 |- ((ph /\ ps /\ ch) <-> ((ph /\ ps) /\ ch))
2		df-3an 583	. . . 4 |- ((th /\ ta /\ et) <-> ((th /\ ta ) /\ et))
3	1, 2	anbi12i 369	. . 3 |- (((ph /\ ps /\ ch) /\ (th /\ ta /\ et)) <-> (((ph /\ ps) /\ ch) /\ ((th /\ ta ) /\ et)))
4		an4 388	. . 3 |- ((((ph /\ ps) /\ ch) /\ ((th /\ ta ) /\ et)) <-> (((ph /\ ps) /\ (th /\ ta )) /\ (ch /\ et)))
5		an4 388	. . . 4 |- (((ph /\ ps) /\ (th /\ ta )) <-> ((ph /\ th) /\ (ps /\ ta )))
6	5	anbi1i 368	. . 3 |- ((((ph /\ ps) /\ (th /\ ta )) /\ (ch /\ et)) <-> (((ph /\ th) /\ (ps /\ ta )) /\ (ch /\ et)))
7	3, 4, 6	3bitr 155	. 2 |- (((ph /\ ps /\ ch) /\ (th /\ ta /\ et)) <-> (((ph /\ th) /\ (ps /\ ta )) /\ (ch /\ et)))
8		df-3an 583	. 2 |- (((ph /\ th) /\ (ps /\ ta ) /\ (ch /\ et)) <-> (((ph /\ th) /\ (ps /\ ta )) /\ (ch /\ et)))
9	7, 8	bitr4 154	1 |- (((ph /\ ps /\ ch) /\ (th /\ ta /\ et)) <-> ((ph /\ th) /\ (ps /\ ta ) /\ (ch /\ et)))

Syntax hints:   <-> wb 127   /\ wa 196   /\ w3a 581

This theorem is referenced by:  f1oco 2816  distrlem3pr 3923

This theorem was proved from axioms:  ax-1 3  ax-2 4  ax-3 5  ax-mp 6

This theorem depends on definitions:  df-bi 128  df-an 198  df-3an 583

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