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  2420  camestres  2703  baroco  2706  rexim  3109  falseral0OLD  4481  nrhmzr  20673  cbvex1v  35494  antnestlaw2  36205  dfon2lem9  36302  hbntg  36316  naim1  36941  naim2  36942  lukshef-ax2  36967  bj-exim  37273  bj-axc10v  37469  ax12indn  39758  cvrexchlem  40234  cvratlem  40236  axfrege28  44596  vk15.4j  45278  tratrb  45286  hbntal  45303  tratrbVD  45610  con5VD  45649  vk15.4jVD  45663