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  1706  nic-axALT  1707  axc10  2417  camestres  2700  baroco  2703  rexim  3107  falseral0OLD  4474  nrhmzr  20822  cbvex1v  35787  antnestlaw2  36637  dfon2lem9  36744  hbntg  36758  naim1  37368  naim2  37369  lukshef-ax2  37394  bj-exim  37700  bj-axc10v  37896  ax12indn  40203  cvrexchlem  40679  cvratlem  40681  axfrege28  45032  vk15.4j  45713  tratrb  45721  hbntal  45738  tratrbVD  46045  con5VD  46084  vk15.4jVD  46098