Theorem con3 156

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 157. (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 155	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  318  nic-ax  1670  nic-axALT  1671  axc10  2399  camestres  2756  baroco  2759  rexim  3241  falseral0  4459  dfon2lem9  33031  hbntg  33045  naim1  33732  naim2  33733  lukshef-ax2  33758  bj-eximALT  33969  bj-axc10v  34110  ax12indn  36073  cvrexchlem  36549  cvratlem  36551  axfrege28  40168  vk15.4j  40855  tratrb  40863  hbntal  40880  tratrbVD  41188  con5VD  41227  vk15.4jVD  41241  nrhmzr  44137