Theorem con3 154

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 155. (Contributed by NM, 29-Dec-1992.) (Proof shortened by Wolf Lammen, 13-Feb-2013.)

Assertion

Ref	Expression
con3	⊢ ((𝜑 → 𝜓) → (¬ 𝜓 → ¬ 𝜑))

Proof of Theorem con3

Step	Hyp	Ref	Expression
1		id 23	. 2 ⊢ ((𝜑 → 𝜓) → (𝜑 → 𝜓))
2	1	con3d 153	1 ⊢ ((𝜑 → 𝜓) → (¬ 𝜓 → ¬ 𝜑))

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  319  nic-ax  1696  nic-axALT  1697  axc10  2419  camestres  2702  baroco  2705  rexim  3106  falseral0OLD  4472  nrhmzr  20610  cbvex1v  35374  antnestlaw2  36050  dfon2lem9  36147  hbntg  36161  naim1  36757  naim2  36758  lukshef-ax2  36783  bj-exim  37089  bj-axc10v  37285  ax12indn  39574  cvrexchlem  40050  cvratlem  40052  axfrege28  44412  vk15.4j  45096  tratrb  45104  hbntal  45121  tratrbVD  45428  con5VD  45467  vk15.4jVD  45481