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  315  nic-ax  1667  nic-axALT  1668  axc10  2379  camestres  2663  baroco  2666  rexim  3083  falseral0  4521  nrhmzr  20479  dfon2lem9  35392  hbntg  35406  naim1  35878  naim2  35879  lukshef-ax2  35904  bj-eximALT  36122  bj-axc10v  36275  ax12indn  38419  cvrexchlem  38896  cvratlem  38898  axfrege28  43262  vk15.4j  43970  tratrb  43978  hbntal  43995  tratrbVD  44303  con5VD  44342  vk15.4jVD  44356