Theorem con2d 627

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

Hypothesis

Ref	Expression
con2d.1	⊢ (𝜑 → (𝜓 → ¬ 𝜒))

Assertion

Ref	Expression
con2d	⊢ (𝜑 → (𝜒 → ¬ 𝜓))

Proof of Theorem con2d

Step	Hyp	Ref	Expression
1		con2d.1	. . . 4 ⊢ (𝜑 → (𝜓 → ¬ 𝜒))
2		ax-in2 618	. . . 4 ⊢ (¬ 𝜒 → (𝜒 → ¬ 𝜓))
3	1, 2	syl6 33	. . 3 ⊢ (𝜑 → (𝜓 → (𝜒 → ¬ 𝜓)))
4	3	com23 78	. 2 ⊢ (𝜑 → (𝜒 → (𝜓 → ¬ 𝜓)))
5		pm2.01 619	. 2 ⊢ ((𝜓 → ¬ 𝜓) → ¬ 𝜓)
6	4, 5	syl6 33	1 ⊢ (𝜑 → (𝜒 → ¬ 𝜓))

Syntax hints:  ¬ wn 3   → wi 4

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-in1 617  ax-in2 618

This theorem is referenced by:  mt2d  628  con3d  634  pm3.2im  640  con2  646  pm2.65  663  con1biimdc  878  exists2  2175  necon2ad  2457  necon2bd  2458  minel  3553  nlimsucg  4658  poirr2  5121  funun  5362  imadif  5401  infnlbti  7193  mkvprop  7325  addnidpig  7523  zltnle  9492  zdcle  9523  btwnnz  9541  prime  9546  icc0r  10122  fznlem  10237  qltnle  10463  bcval4  10974  seq3coll  11064  swrd0g  11192  fsum3cvg  11889  fsumsplit  11918  fproddccvg  12083  fprodsplitdc  12107  bitsinv1lem  12472  2sqpwodd  12698  pockthg  12880  prmunb  12885  logbgcd1irr  15641  lgsne0  15717