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  9694  zdcle  9725  btwnnz  9744  prime  9749  icc0r  10338  fznlem  10455  qltnle  10688  bcval4  11204  hashf1  11301  seq3coll  11308  swrd0g  11446  fsum3cvg  12161  fsumsplit  12190  fproddccvg  12355  fprodsplitdc  12379  bitsinv1lem  12744  2sqpwodd  12972  pockthg  13156  prmunb  13161  ballotfilemfc0  13281  ballotfilemfcc  13282  ballotfilemirc  13324  logbgcd1irr  16122  lgsne0  16255  eupth2lem3lem4fi  16812