Theorem pm2.53 674

Description: Theorem *2.53 of [WhiteheadRussell] p. 107. This holds intuitionistically, although its converse does not (see pm2.54dc 824). (Contributed by NM, 3-Jan-2005.) (Revised by NM, 31-Jan-2015.)

Assertion

Ref	Expression
pm2.53	⊢ ((𝜑 ∨ 𝜓) → (¬ 𝜑 → 𝜓))

Proof of Theorem pm2.53

Step	Hyp	Ref	Expression
1		pm2.24 584	. 2 ⊢ (𝜑 → (¬ 𝜑 → 𝜓))
2		ax-1 5	. 2 ⊢ (𝜓 → (¬ 𝜑 → 𝜓))
3	1, 2	jaoi 669	1 ⊢ ((𝜑 ∨ 𝜓) → (¬ 𝜑 → 𝜓))

Syntax hints:  ¬ wn 3   → wi 4   ∨ wo 662

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106  ax-in2 578  ax-io 663

This theorem depends on definitions:  df-bi 115

This theorem is referenced by:  ori  675  ord  676  orel1  677  pm2.63  747  notnotrdc  785  dfordc  825  pm5.6r  870  xorbin  1316  19.33b2  1561  onsucelsucexmid  4308  oprabidlem  5614  xnn0nnn0pnf  8642  absle  10348