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  192  con34b  316  nic-ax  1680  nic-axALT  1681  axc10  2387  camestres  2676  baroco  2679  rexim  3171  falseral0  4456  dfon2lem9  33776  hbntg  33790  naim1  34587  naim2  34588  lukshef-ax2  34613  bj-eximALT  34831  bj-axc10v  34984  ax12indn  36966  cvrexchlem  37442  cvratlem  37444  axfrege28  41419  vk15.4j  42130  tratrb  42138  hbntal  42155  tratrbVD  42463  con5VD  42502  vk15.4jVD  42516  nrhmzr  45410