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  2390  camestres  2673  baroco  2676  rexim  3078  falseral0  4496  nrhmzr  20502  cbvex1v  35110  dfon2lem9  35814  hbntg  35828  naim1  36412  naim2  36413  lukshef-ax2  36438  bj-eximALT  36664  bj-axc10v  36816  ax12indn  38966  cvrexchlem  39443  cvratlem  39445  axfrege28  43820  vk15.4j  44520  tratrb  44528  hbntal  44545  tratrbVD  44852  con5VD  44891  vk15.4jVD  44905