Theorem con3d 640

Description: A contraposition deduction. (Contributed by NM, 5-Aug-1993.) (Revised by NM, 31-Jan-2015.)

Hypothesis

Ref	Expression
con3d.1	|- ( ph -> ( ps -> ch ) )

Assertion

Ref	Expression
con3d	|- ( ph -> ( -. ch -> -. ps ) )

Proof of Theorem con3d

Step	Hyp	Ref	Expression
1		con3d.1	. . 3 |- ( ph -> ( ps -> ch ) )
2		notnot 638	. . 3 |- ( ch -> -. -. ch )
3	1, 2	syl6 33	. 2 |- ( ph -> ( ps -> -. -. ch ) )
4	3	con2d 633	1 |- ( ph -> ( -. ch -> -. ps ) )

This proof depends on syntax axioms:   -. wn 3   -> wi 4

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-in1 623  ax-in2 624

This theorem is used by:  con3rr3  642  con3dimp  644  con3  651  nsyld  657  nsyli  658  jcn  661  notbi  676  impidc  870  bijadc  894  pm2.13dc  897  xoranor  1426  mo2n  2114  necon3ad  2462  necon3bd  2463  nelcon3d  2526  ssneld  3250  sscon  3363  difrab  3507  exmid1stab  4345  eunex  4708  ndmfvg  5726  suppssrst  6501  suppssrgst  6502  nnaord  6782  nnmord  6790  php5  7159  php5dom  7164  fidcen  7203  supmoti  7333  exmidomniim  7481  mkvprop  7498  enmkvlem  7501  prubl  7853  letr  8408  eqord1  8812  prodge0  9186  lt2msq  9218  nnge1  9329  nzadd  9701  irradd  10055  irrmul  10057  xrletr  10220  frec2uzf1od  10856  zesq  11109  expcanlem  11167  nn0opthd  11174  bccmpl  11206  fundm2domnop0  11314  maxleast  11994  fisumss  12175  dvdsbnd  12749  prm2orodd  12920  coprm  12939  prmndvdsfaclt  12951  nn0sqdcq  13004  hashgcdeq  13038  ballotfilemfc0  13281  ballotfilemfcc  13282  cos11  16004  logdivlt  16046  bposlem3  16211  bj-nnsn  16859  bj-nnelirr  17077  ismkvnnlem  17200  nconstwlpolem  17213