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))

This proof depends on syntax axioms:  ¬ wn 3   → wi 4

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8

This theorem is used by:  pm2.65  195  con34b  319  nic-ax  1705  nic-axALT  1706  axc10  2414  camestres  2697  baroco  2700  rexim  3103  falseral0OLD  4470  nrhmzr  20795  cbvex1v  35751  antnestlaw2  36544  dfon2lem9  36646  hbntg  36660  naim1  37270  naim2  37271  lukshef-ax2  37296  bj-exim  37602  bj-axc10v  37798  ax12indn  40094  cvrexchlem  40570  cvratlem  40572  axfrege28  44923  vk15.4j  45605  tratrb  45613  hbntal  45630  tratrbVD  45937  con5VD  45976  vk15.4jVD  45990