Metamath Proof Explorer < Previous   Next > Nearby theorems Mirrors  >  Home  >  MPE Home  >  Th. List  >  con3 Structured version   Visualization version   GIF version

Theorem con3 149
 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 150. (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 148 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  184  con34b  305  nic-ax  1638  nic-axALT  1639  axc10  2288  rexim  3037  ralf0OLD  4112  falseral0  4114  dfon2lem9  31820  hbntg  31835  naim1  32509  naim2  32510  lukshef-ax2  32539  bj-axc10v  32842  ax12indn  34547  cvrexchlem  35023  cvratlem  35025  axfrege28  38440  vk15.4j  39051  tratrb  39063  hbntal  39086  tratrbVD  39411  con5VD  39450  vk15.4jVD  39464  nrhmzr  42198
 Copyright terms: Public domain W3C validator