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  7366  mkvprop  7498  addnidpig  7703  zltnle  9690  zdcle  9721  btwnnz  9740  prime  9745  icc0r  10328  fznlem  10445  qltnle  10678  bcval4  11190  hashf1  11287  seq3coll  11294  swrd0g  11432  fsum3cvg  12145  fsumsplit  12174  fproddccvg  12339  fprodsplitdc  12363  bitsinv1lem  12728  2sqpwodd  12954  pockthg  13136  prmunb  13141  ballotfilemfc0  13232  ballotfilemfcc  13233  ballotfilemirc  13275  logbgcd1irr  16069  lgsne0  16157  eupth2lem3lem4fi  16714