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  979  an6  1334  2eu4  2147  reean  2675  reu2  2961  rmo4  2966  rmo3f  2970  rmo3  3090  inxp  4812  xp11m  5121  fununi  5342  fun  5448  resoprab2  6042  xporderlem  6317  poxp  6318  th3qlem1  6724  enq0enq  7544  enq0tr  7547  genpdisj  7636  cju  9034  elfzo2  10272  iooinsup  11588  summodc  11694  prodmodc  11889  issubmd  13306  dvdsrtr  13863  domnmuln0  14035  txbasval  14739  txcnp  14743  txlm  14751