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  8811  prodge0  9184  lt2msq  9216  nnge1  9327  nzadd  9697  irradd  10046  irrmul  10047  xrletr  10210  frec2uzf1od  10843  zesq  11096  expcanlem  11153  nn0opthd  11160  bccmpl  11192  fundm2domnop0  11300  maxleast  11979  fisumss  12159  dvdsbnd  12733  prm2orodd  12904  coprm  12922  prmndvdsfaclt  12934  hashgcdeq  13018  ballotfilemfc0  13232  ballotfilemfcc  13233  cos11  15954  bj-nnsn  16761  bj-nnelirr  16979  ismkvnnlem  17102  nconstwlpolem  17115