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Theorem con3 147
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 148. (Contributed by NM, 29-Dec-1992.) (Proof shortened by Wolf Lammen, 13-Feb-2013.)
Assertion
Ref Expression
con3 ((𝜑𝜓) → (¬ 𝜓 → ¬ 𝜑))

Proof of Theorem con3
StepHypRef Expression
1 id 22 . 2 ((𝜑𝜓) → (𝜑𝜓))
21con3d 146 1 ((𝜑𝜓) → (¬ 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  182  con34b  304  nic-ax  1588  nic-axALT  1589  axc10  2235  rexim  2987  ralf0OLD  4027  dfon2lem9  30743  hbntg  30758  naim1  31357  naim2  31358  lukshef-ax2  31387  bj-axc10v  31707  ax12indn  33046  cvrexchlem  33523  cvratlem  33525  axfrege28  36943  vk15.4j  37555  tratrb  37567  hbntal  37590  tratrbVD  37919  con5VD  37958  vk15.4jVD  37972  falseral0  40110  nrhmzr  41662
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