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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 157. (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 155 | 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: pm2.65 195 con34b 318 nic-ax 1674 nic-axALT 1675 axc10 2403 camestres 2758 baroco 2761 rexim 3241 falseral0 4459 dfon2lem9 33036 hbntg 33050 naim1 33737 naim2 33738 lukshef-ax2 33763 bj-eximALT 33974 bj-axc10v 34115 ax12indn 36094 cvrexchlem 36570 cvratlem 36572 axfrege28 40195 vk15.4j 40882 tratrb 40890 hbntal 40907 tratrbVD 41215 con5VD 41254 vk15.4jVD 41268 nrhmzr 44164 |
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