Theorem con3 153

Description: Contraposition. Theorem *2.16 of [WhiteheadRussell] p. 103. This was the fourth axiom of Frege, specifically Proposition 28 of [Frege1879] p. 43. Its associated inference is con3i 154. (Contributed by NM, 29-Dec-1992.) (Proof shortened by Wolf Lammen, 13-Feb-2013.)

Assertion

Ref	Expression
con3	⊢ ((𝜑 → 𝜓) → (¬ 𝜓 → ¬ 𝜑))

Proof of Theorem con3

Step	Hyp	Ref	Expression
1		id 22	. 2 ⊢ ((𝜑 → 𝜓) → (𝜑 → 𝜓))
2	1	con3d 152	1 ⊢ ((𝜑 → 𝜓) → (¬ 𝜓 → ¬ 𝜑))

Syntax hints:  ¬ wn 3   → wi 4

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8

This theorem is referenced by:  pm2.65  194  con34b  318  nic-ax  1692  nic-axALT  1693  axc10  2415  camestres  2698  baroco  2701  rexim  3102  falseral0OLD  4466  nrhmzr  20573  cbvex1v  35329  antnestlaw2  36002  dfon2lem9  36099  hbntg  36113  naim1  36709  naim2  36710  lukshef-ax2  36735  bj-exim  37042  bj-axc10v  37238  ax12indn  39527  cvrexchlem  40003  cvratlem  40005  axfrege28  44365  vk15.4j  45064  tratrb  45072  hbntal  45089  tratrbVD  45396  con5VD  45435  vk15.4jVD  45449