Theorem con3 151

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 152. (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 150	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  185  con34b  308  nic-ax  1772  nic-axALT  1773  axc10  2404  camestres  2757  baroco  2761  rexim  3216  falseral0  4301  dfon2lem9  32223  hbntg  32238  naim1  32911  naim2  32912  lukshef-ax2  32936  bj-axc10v  33244  ax12indn  35011  cvrexchlem  35487  cvratlem  35489  axfrege28  38956  vk15.4j  39565  tratrb  39573  hbntal  39590  tratrbVD  39908  con5VD  39947  vk15.4jVD  39961  nrhmzr  42713