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  4347  eunex  4710  ndmfvg  5729  suppssrst  6505  suppssrgst  6506  nnaord  6787  nnmord  6795  php5  7165  php5dom  7170  fidcen  7209  supmoti  7339  exmidomniim  7487  mkvprop  7504  enmkvlem  7507  prubl  7859  letr  8414  eqord1  8818  prodge0  9192  lt2msq  9224  nnge1  9336  nzadd  9706  irradd  10061  irrmul  10063  xrletr  10226  frec2uzf1od  10863  zesq  11116  expcanlem  11174  nn0opthd  11181  bccmpl  11213  fundm2domnop0  11321  maxleast  12001  fisumss  12183  dvdsbnd  12757  prm2orodd  12928  coprm  12947  prmndvdsfaclt  12959  nn0sqdcq  13012  hashgcdeq  13046  ballotfilemfc0  13289  ballotfilemfcc  13290  cos11  16098  logdivlt  16141  bposlem3  16330  bj-nnsn  16983  bj-nnelirr  17201  ismkvnnlem  17326  nconstwlpolem  17339