Theorem orel2 645

Description: Elimination of disjunction by denial of a disjunct. Theorem *2.56 of [WhiteheadRussell] p. 107. (Contributed by NM, 12-Aug-1994.) (Proof shortened by Wolf Lammen, 5-Apr-2013.)

Assertion

Ref	Expression
orel2	⊢ (¬ 𝜑 → ((𝜓 ∨ 𝜑) → 𝜓))

Proof of Theorem orel2

Step	Hyp	Ref	Expression
1		idd 21	. 2 ⊢ (¬ 𝜑 → (𝜓 → 𝜓))
2		pm2.21 547	. 2 ⊢ (¬ 𝜑 → (𝜑 → 𝜓))
3	1, 2	jaod 637	1 ⊢ (¬ 𝜑 → ((𝜓 ∨ 𝜑) → 𝜓))

Syntax hints:  ¬ wn 3   → wi 4   ∨ wo 629

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 99  ax-ia2 100  ax-ia3 101  ax-in2 545  ax-io 630

This theorem depends on definitions:  df-bi 110

This theorem is referenced by:  biorfi  665  pm2.64  714  pm5.71dc  868  ecased  1239  19.30dc  1518  dveeq2  1696  prel12  3542  funun  4944