Theorem an6 900

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 776	. . . 4 |- ((ph /\ ps /\ ch) <-> ((ph /\ ps) /\ ch))
2		df-3an 776	. . . 4 |- ((th /\ ta /\ et) <-> ((th /\ ta) /\ et))
3	1, 2	anbi12i 482	. . 3 |- (((ph /\ ps /\ ch) /\ (th /\ ta /\ et)) <-> (((ph /\ ps) /\ ch) /\ ((th /\ ta) /\ et)))
4		an4 506	. . 3 |- ((((ph /\ ps) /\ ch) /\ ((th /\ ta) /\ et)) <-> (((ph /\ ps) /\ (th /\ ta)) /\ (ch /\ et)))
5		an4 506	. . . 4 |- (((ph /\ ps) /\ (th /\ ta)) <-> ((ph /\ th) /\ (ps /\ ta)))
6	5	anbi1i 481	. . 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 776	. 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 774

This theorem is referenced by:  abfii4 4544  distrlem3pr 5109  ltdiv2t 5843  elfzuzb 6416  efcltlem1 7254  subbas 7594  iscau3 7890  iscau4 7892  infi1 10383  ficli 10404  filintf 10479  infi 10484

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 776

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