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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 149. (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 147 | 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: pm2.65 183 con34b 305 nic-ax 1589 nic-axALT 1590 axc10 2240 rexim 2991 ralf0OLD 4031 dfon2lem9 30940 hbntg 30955 naim1 31554 naim2 31555 lukshef-ax2 31584 bj-axc10v 31904 ax12indn 33246 cvrexchlem 33723 cvratlem 33725 axfrege28 37143 vk15.4j 37755 tratrb 37767 hbntal 37790 tratrbVD 38119 con5VD 38158 vk15.4jVD 38172 falseral0 40308 nrhmzr 41663 |
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