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| 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.) | 
| Ref | Expression | 
|---|---|
| con3 | ⊢ ((𝜑 → 𝜓) → (¬ 𝜓 → ¬ 𝜑)) | 
| Step | Hyp | Ref | Expression | 
|---|---|---|---|
| 1 | id 22 | . 2 ⊢ ((𝜑 → 𝜓) → (𝜑 → 𝜓)) | |
| 2 | 1 | con3d 152 | 1 ⊢ ((𝜑 → 𝜓) → (¬ 𝜓 → ¬ 𝜑)) | 
| Colors of variables: wff setvar class | 
| 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: pm2.65 193 con34b 316 nic-ax 1672 nic-axALT 1673 axc10 2389 camestres 2672 baroco 2675 rexim 3086 falseral0 4515 nrhmzr 20538 cbvex1v 35089 dfon2lem9 35793 hbntg 35807 naim1 36391 naim2 36392 lukshef-ax2 36417 bj-eximALT 36643 bj-axc10v 36795 ax12indn 38945 cvrexchlem 39422 cvratlem 39424 axfrege28 43847 vk15.4j 44553 tratrb 44561 hbntal 44578 tratrbVD 44886 con5VD 44925 vk15.4jVD 44939 | 
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