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  978  an6  1332  2eu4  2138  reean  2666  reu2  2952  rmo4  2957  rmo3f  2961  rmo3  3081  inxp  4800  xp11m  5108  fununi  5326  fun  5430  resoprab2  6019  xporderlem  6289  poxp  6290  th3qlem1  6696  enq0enq  7498  enq0tr  7501  genpdisj  7590  cju  8988  elfzo2  10225  iooinsup  11442  summodc  11548  prodmodc  11743  issubmd  13106  dvdsrtr  13657  domnmuln0  13829  txbasval  14503  txcnp  14507  txlm  14515