Theorem pm2.53 730

Description: Theorem *2.53 of [WhiteheadRussell] p. 107. This holds intuitionistically, although its converse does not (see pm2.54dc 899). (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		pm2.24 626	. 2 ⊢ (𝜑 → (¬ 𝜑 → 𝜓))
2		ax-1 6	. 2 ⊢ (𝜓 → (¬ 𝜑 → 𝜓))
3	1, 2	jaoi 724	1 ⊢ ((𝜑 ∨ 𝜓) → (¬ 𝜑 → 𝜓))

Syntax hints:  ¬ wn 3   → wi 4   ∨ wo 716

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-in2 620  ax-io 717

This theorem depends on definitions:  df-bi 117

This theorem is referenced by:  ori  731  ord  732  orel1  733  pm2.63  808  notnotrdc  851  dfordc  900  pm5.6r  935  xorbin  1429  19.33b2  1678  r19.30dc  2681  onsucelsucexmid  4634  oprabidlem  6059  omnimkv  7398  xnn0nnn0pnf  9522  absle  11712