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  7334  exmidomniim  7482  mkvprop  7499  enmkvlem  7502  prubl  7854  letr  8409  eqord1  8813  prodge0  9187  lt2msq  9219  nnge1  9330  nzadd  9702  irradd  10056  irrmul  10058  xrletr  10221  frec2uzf1od  10858  zesq  11111  expcanlem  11169  nn0opthd  11176  bccmpl  11208  fundm2domnop0  11316  maxleast  11996  fisumss  12178  dvdsbnd  12752  prm2orodd  12923  coprm  12942  prmndvdsfaclt  12954  nn0sqdcq  13007  hashgcdeq  13041  ballotfilemfc0  13284  ballotfilemfcc  13285  cos11  16046  logdivlt  16088  bposlem3  16274  bj-nnsn  16927  bj-nnelirr  17145  ismkvnnlem  17269  nconstwlpolem  17282