Theorem pm2.53 712

Description: Theorem *2.53 of [WhiteheadRussell] p. 107. This holds intuitionistically, although its converse does not (see pm2.54dc 877). (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		pm2.24 611	. 2 ⊢ (𝜑 → (¬ 𝜑 → 𝜓))
2		ax-1 6	. 2 ⊢ (𝜓 → (¬ 𝜑 → 𝜓))
3	1, 2	jaoi 706	1 ⊢ ((𝜑 ∨ 𝜓) → (¬ 𝜑 → 𝜓))

Syntax hints:  ¬ wn 3   → wi 4   ∨ wo 698

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-in2 605  ax-io 699

This theorem depends on definitions:  df-bi 116

This theorem is referenced by:  ori  713  ord  714  orel1  715  pm2.63  790  notnotrdc  829  dfordc  878  pm5.6r  913  xorbin  1366  19.33b2  1609  r19.30dc  2604  onsucelsucexmid  4489  oprabidlem  5852  omnimkv  7099  xnn0nnn0pnf  9166  absle  10989