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  2389  camestres  2673  baroco  2676  rexim  3078  falseral0OLD  4455  nrhmzr  20514  cbvex1v  35216  antnestlaw2  35874  dfon2lem9  35971  hbntg  35985  naim1  36571  naim2  36572  lukshef-ax2  36597  bj-exim  36904  bj-axc10v  37100  ax12indn  39389  cvrexchlem  39865  cvratlem  39867  axfrege28  44256  vk15.4j  44955  tratrb  44963  hbntal  44980  tratrbVD  45287  con5VD  45326  vk15.4jVD  45340