Theorem 3com12 1238

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 1016	. 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 )

This proof depends on syntax axioms:   -> wi 4   /\ w3a 1009

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108

This proof depends on definitions:  df-bi 117  df-3an 1011

This theorem is used by:  3adant2l  1263  3adant2r  1264  brelrng  5013  iotam  5369  funimaexglem  5464  fresaunres1disj  5571  fvun2  5770  nnaordi  6781  nnmword  6791  fpmg  6955  prcdnql  7851  prcunqu  7852  prarloc  7870  ltaprg  7986  mul12  8455  add12  8484  addsub  8537  addsubeq4  8541  ppncan  8568  leadd1  8758  ltaddsub2  8765  leaddsub2  8767  lemul1  8921  reapmul1lem  8922  reapadd1  8924  reapcotr  8926  remulext1  8927  div23ap  9021  ltmulgt11  9194  lediv1  9199  lemuldiv  9211  zdiv  9734  iooneg  10390  icoshft  10392  fzaddel  10465  fzshftral  10515  facwordi  11178  pfxeq  11468  abssubge0  11868  climshftlemg  12068  dvdsmul1  12580  divalgb  12692  lcmgcdeq  12861  pcfac  13129  mhmmulg  13966  rmodislmodlem  14687  cnmptcom  15399  hmeof1o2  15409