Theorem an4 586

Description: Rearrangement of 4 conjuncts. (Contributed by NM, 10-Jul-1994.)

Assertion

Ref	Expression
an4	|- ( ( ( ph /\ ps ) /\ ( ch /\ th ) ) <-> ( ( ph /\ ch ) /\ ( ps /\ th ) ) )

Proof of Theorem an4

Step	Hyp	Ref	Expression
1		an12 561	. . 3 |- ( ( ps /\ ( ch /\ th ) ) <-> ( ch /\ ( ps /\ th ) ) )
2	1	anbi2i 457	. 2 |- ( ( ph /\ ( ps /\ ( ch /\ th ) ) ) <-> ( ph /\ ( ch /\ ( ps /\ th ) ) ) )
3		anass 401	. 2 |- ( ( ( ph /\ ps ) /\ ( ch /\ th ) ) <-> ( ph /\ ( ps /\ ( ch /\ th ) ) ) )
4		anass 401	. 2 |- ( ( ( ph /\ ch ) /\ ( ps /\ th ) ) <-> ( ph /\ ( ch /\ ( ps /\ th ) ) ) )
5	2, 3, 4	3bitr4i 212	1 |- ( ( ( ph /\ ps ) /\ ( ch /\ th ) ) <-> ( ( ph /\ ch ) /\ ( ps /\ th ) ) )

Syntax hints:   /\ wa 104   <-> wb 105

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108

This theorem depends on definitions:  df-bi 117

This theorem is referenced by:  an42  587  an4s  588  anandi  590  anandir  591  rnlem  976  an6  1321  2eu4  2119  reean  2645  reu2  2925  rmo4  2930  rmo3f  2934  rmo3  3054  inxp  4761  xp11m  5067  fununi  5284  fun  5388  resoprab2  5971  xporderlem  6231  poxp  6232  th3qlem1  6636  enq0enq  7429  enq0tr  7432  genpdisj  7521  cju  8916  elfzo2  10147  iooinsup  11280  summodc  11386  prodmodc  11581  issubmd  12859  dvdsrtr  13263  txbasval  13698  txcnp  13702  txlm  13710