Theorem 3com12 1231

Description: Commutation in antecedent. Swap 1st and 3rd. (Contributed by NM, 28-Jan-1996.) (Proof shortened by Andrew Salmon, 13-May-2011.)

Hypothesis

Ref	Expression
3exp.1	|- ( ( ph /\ ps /\ ch ) -> th )

Assertion

Ref	Expression
3com12	|- ( ( ps /\ ph /\ ch ) -> th )

Proof of Theorem 3com12

Step	Hyp	Ref	Expression
1		3ancoma 1009	. 2 |- ( ( ps /\ ph /\ ch ) <-> ( ph /\ ps /\ ch ) )
2		3exp.1	. 2 |- ( ( ph /\ ps /\ ch ) -> th )
3	1, 2	sylbi 121	1 |- ( ( ps /\ ph /\ ch ) -> th )

Syntax hints:   -> wi 4   /\ w3a 1002

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  df-3an 1004

This theorem is referenced by:  3adant2l  1256  3adant2r  1257  brelrng  4955  iotam  5310  funimaexglem  5404  fvun2  5701  nnaordi  6654  nnmword  6664  fpmg  6821  prcdnql  7671  prcunqu  7672  prarloc  7690  ltaprg  7806  mul12  8275  add12  8304  addsub  8357  addsubeq4  8361  ppncan  8388  leadd1  8577  ltaddsub2  8584  leaddsub2  8586  lemul1  8740  reapmul1lem  8741  reapadd1  8743  reapcotr  8745  remulext1  8746  div23ap  8838  ltmulgt11  9011  lediv1  9016  lemuldiv  9028  zdiv  9535  iooneg  10184  icoshft  10186  fzaddel  10255  fzshftral  10304  facwordi  10962  pfxeq  11228  abssubge0  11613  climshftlemg  11813  dvdsmul1  12324  divalgb  12436  lcmgcdeq  12605  pcfac  12873  mhmmulg  13700  rmodislmodlem  14314  cnmptcom  14972  hmeof1o2  14982