Theorem con3 153

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 154. (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 22	. 2 ⊢ ((𝜑 → 𝜓) → (𝜑 → 𝜓))
2	1	con3d 152	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  194  con34b  317  nic-ax  1680  nic-axALT  1681  axc10  2393  camestres  2677  baroco  2680  rexim  3081  falseral0OLD  4450  nrhmzr  20516  cbvex1v  35263  antnestlaw2  35927  dfon2lem9  36024  hbntg  36038  naim1  36624  naim2  36625  lukshef-ax2  36650  bj-exim  36957  bj-axc10v  37153  ax12indn  39442  cvrexchlem  39918  cvratlem  39920  axfrege28  44280  vk15.4j  44979  tratrb  44987  hbntal  45004  tratrbVD  45311  con5VD  45350  vk15.4jVD  45364