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Mirrors > Home > MPE Home > Th. List > con3 | Structured version Visualization version GIF version |
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 152. (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 150 | 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: pm2.65 185 con34b 308 nic-ax 1772 nic-axALT 1773 axc10 2404 camestres 2757 baroco 2761 rexim 3216 falseral0 4301 dfon2lem9 32223 hbntg 32238 naim1 32911 naim2 32912 lukshef-ax2 32936 bj-axc10v 33244 ax12indn 35011 cvrexchlem 35487 cvratlem 35489 axfrege28 38956 vk15.4j 39565 tratrb 39573 hbntal 39590 tratrbVD 39908 con5VD 39947 vk15.4jVD 39961 nrhmzr 42713 |
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