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  1674  nic-axALT  1675  axc10  2387  camestres  2670  baroco  2673  rexim  3074  falseral0OLD  4465  nrhmzr  20456  cbvex1v  35109  antnestlaw2  35759  dfon2lem9  35856  hbntg  35870  naim1  36456  naim2  36457  lukshef-ax2  36482  bj-eximALT  36708  bj-axc10v  36860  ax12indn  39065  cvrexchlem  39541  cvratlem  39543  axfrege28  43949  vk15.4j  44648  tratrb  44656  hbntal  44673  tratrbVD  44980  con5VD  45019  vk15.4jVD  45033