Theorem pm2.53 734

Description: Theorem *2.53 of [WhiteheadRussell] p. 107. This holds intuitionistically, although its converse does not (see pm2.54dc 903). (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		pm2.24 630	. 2 ⊢ (𝜑 → (¬ 𝜑 → 𝜓))
2		ax-1 6	. 2 ⊢ (𝜓 → (¬ 𝜑 → 𝜓))
3	1, 2	jaoi 728	1 ⊢ ((𝜑 ∨ 𝜓) → (¬ 𝜑 → 𝜓))

This proof depends on syntax axioms:  ¬ wn 3   → wi 4   ∨ wo 720

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-in2 624  ax-io 721

This proof depends on definitions:  df-bi 117

This theorem is used by:  ori  735  ord  736  orel1  737  pm2.63  812  notnotrdc  855  dfordc  904  pm5.6r  939  xorbin  1433  19.33b2  1682  r19.30dc  2698  onsucelsucexmid  4677  oprabidlem  6116  omnimkv  7496  xnn0nnn0pnf  9643  absle  11855