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  1675  nic-axALT  1676  axc10  2390  camestres  2674  baroco  2677  rexim  3079  falseral0OLD  4456  nrhmzr  20505  cbvex1v  35232  antnestlaw2  35890  dfon2lem9  35987  hbntg  36001  naim1  36587  naim2  36588  lukshef-ax2  36613  bj-exim  36920  bj-axc10v  37116  ax12indn  39403  cvrexchlem  39879  cvratlem  39881  axfrege28  44274  vk15.4j  44973  tratrb  44981  hbntal  44998  tratrbVD  45305  con5VD  45344  vk15.4jVD  45358