Theorem an6 899

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 775	. . . 4 |- ((ph /\ ps /\ ch) <-> ((ph /\ ps) /\ ch))
2		df-3an 775	. . . 4 |- ((th /\ ta /\ et) <-> ((th /\ ta) /\ et))
3	1, 2	anbi12i 481	. . 3 |- (((ph /\ ps /\ ch) /\ (th /\ ta /\ et)) <-> (((ph /\ ps) /\ ch) /\ ((th /\ ta) /\ et)))
4		an4 505	. . 3 |- ((((ph /\ ps) /\ ch) /\ ((th /\ ta) /\ et)) <-> (((ph /\ ps) /\ (th /\ ta)) /\ (ch /\ et)))
5		an4 505	. . . 4 |- (((ph /\ ps) /\ (th /\ ta)) <-> ((ph /\ th) /\ (ps /\ ta)))
6	5	anbi1i 480	. . 3 |- ((((ph /\ ps) /\ (th /\ ta)) /\ (ch /\ et)) <-> (((ph /\ th) /\ (ps /\ ta)) /\ (ch /\ et)))
7	3, 4, 6	3bitr 177	. 2 |- (((ph /\ ps /\ ch) /\ (th /\ ta /\ et)) <-> (((ph /\ th) /\ (ps /\ ta)) /\ (ch /\ et)))
8		df-3an 775	. 2 |- (((ph /\ th) /\ (ps /\ ta) /\ (ch /\ et)) <-> (((ph /\ th) /\ (ps /\ ta)) /\ (ch /\ et)))
9	7, 8	bitr4 176	1 |- (((ph /\ ps /\ ch) /\ (th /\ ta /\ et)) <-> ((ph /\ th) /\ (ps /\ ta) /\ (ch /\ et)))

Syntax hints:   <-> wb 146   /\ wa 223   /\ w3a 773

This theorem is referenced by:  abfii4 4538  distrlem3pr 5101  ltdiv2t 5835  elfzuzb 6408  efcltlem1 7246  subbas 7586  iscau3 7876  infi1 10347  ficli 10368  filintf 10443  infi 10448

This theorem was proved from axioms:  ax-1 4  ax-2 5  ax-3 6  ax-mp 7

This theorem depends on definitions:  df-bi 147  df-an 225  df-3an 775

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