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  1671  nic-axALT  1672  axc10  2393  camestres  2676  baroco  2679  rexim  3093  falseral0  4539  nrhmzr  20563  cbvex1v  35050  dfon2lem9  35755  hbntg  35769  naim1  36355  naim2  36356  lukshef-ax2  36381  bj-eximALT  36607  bj-axc10v  36759  ax12indn  38899  cvrexchlem  39376  cvratlem  39378  axfrege28  43791  vk15.4j  44499  tratrb  44507  hbntal  44524  tratrbVD  44832  con5VD  44871  vk15.4jVD  44885