Theorem con2d 633

Description: A contraposition deduction. (Contributed by NM, 19-Aug-1993.) (Revised by NM, 12-Feb-2013.)

Hypothesis

Ref	Expression
con2d.1	|- ( ph -> ( ps -> -. ch ) )

Assertion

Ref	Expression
con2d	|- ( ph -> ( ch -> -. ps ) )

Proof of Theorem con2d

Step	Hyp	Ref	Expression
1		con2d.1	. . . 4 |- ( ph -> ( ps -> -. ch ) )
2		ax-in2 624	. . . 4 |- ( -. ch -> ( ch -> -. ps ) )
3	1, 2	syl6 33	. . 3 |- ( ph -> ( ps -> ( ch -> -. ps ) ) )
4	3	com23 78	. 2 |- ( ph -> ( ch -> ( ps -> -. ps ) ) )
5		pm2.01 625	. 2 |- ( ( ps -> -. ps ) -> -. ps )
6	4, 5	syl6 33	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:  mt2d  634  con3d  640  pm3.2im  646  con2  652  pm2.65  669  con1biimdc  885  exists2  2184  necon2ad  2477  necon2bd  2478  minel  3586  nlimsucg  4713  poirr2  5180  funun  5422  imadif  5461  infnlbti  7367  mkvprop  7499  addnidpig  7704  zltnle  9695  zdcle  9726  btwnnz  9745  prime  9750  icc0r  10339  fznlem  10456  qltnle  10689  bcval4  11206  hashf1  11303  seq3coll  11310  swrd0g  11448  fsum3cvg  12164  fsumsplit  12193  fproddccvg  12358  fprodsplitdc  12382  bitsinv1lem  12747  2sqpwodd  12975  pockthg  13159  prmunb  13164  ballotfilemfc0  13284  ballotfilemfcc  13285  ballotfilemirc  13327  logbgcd1irr  16164  chtqub  16257  lgsne0  16323  eupth2lem3lem4fi  16880