Theorem con3 156

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 157. (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 155	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  196  con34b  319  nic-ax  1675  nic-axALT  1676  axc10  2392  camestres  2735  baroco  2738  rexim  3204  falseral0  4417  dfon2lem9  33149  hbntg  33163  naim1  33850  naim2  33851  lukshef-ax2  33876  bj-eximALT  34087  bj-axc10v  34230  ax12indn  36239  cvrexchlem  36715  cvratlem  36717  axfrege28  40530  vk15.4j  41234  tratrb  41242  hbntal  41259  tratrbVD  41567  con5VD  41606  vk15.4jVD  41620  nrhmzr  44497