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  1676  nic-axALT  1677  axc10  2385  camestres  2669  baroco  2672  rexim  3088  falseral0  4520  dfon2lem9  34763  hbntg  34777  naim1  35274  naim2  35275  lukshef-ax2  35300  bj-eximALT  35518  bj-axc10v  35671  ax12indn  37813  cvrexchlem  38290  cvratlem  38292  axfrege28  42580  vk15.4j  43289  tratrb  43297  hbntal  43314  tratrbVD  43622  con5VD  43661  vk15.4jVD  43675  nrhmzr  46647