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  193  con34b  316  nic-ax  1673  nic-axALT  1674  axc10  2383  camestres  2666  baroco  2669  rexim  3070  falseral0  4469  nrhmzr  20440  cbvex1v  35043  antnestlaw2  35667  dfon2lem9  35767  hbntg  35781  naim1  36365  naim2  36366  lukshef-ax2  36391  bj-eximALT  36617  bj-axc10v  36769  ax12indn  38924  cvrexchlem  39401  cvratlem  39403  axfrege28  43805  vk15.4j  44505  tratrb  44513  hbntal  44530  tratrbVD  44837  con5VD  44876  vk15.4jVD  44890