Theorem con3 148

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 149. (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 147	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  183  con34b  305  nic-ax  1589  nic-axALT  1590  axc10  2240  rexim  2991  ralf0OLD  4031  dfon2lem9  30940  hbntg  30955  naim1  31554  naim2  31555  lukshef-ax2  31584  bj-axc10v  31904  ax12indn  33246  cvrexchlem  33723  cvratlem  33725  axfrege28  37143  vk15.4j  37755  tratrb  37767  hbntal  37790  tratrbVD  38119  con5VD  38158  vk15.4jVD  38172  falseral0  40308  nrhmzr  41663