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  1675  nic-axALT  1676  axc10  2384  camestres  2668  baroco  2671  rexim  3087  falseral0  4519  dfon2lem9  34758  hbntg  34772  naim1  35269  naim2  35270  lukshef-ax2  35295  bj-eximALT  35513  bj-axc10v  35666  ax12indn  37808  cvrexchlem  38285  cvratlem  38287  axfrege28  42570  vk15.4j  43279  tratrb  43287  hbntal  43304  tratrbVD  43612  con5VD  43651  vk15.4jVD  43665  nrhmzr  46637