Theorem ornld 1059

Description: Selecting one statement from a disjunction if one of the disjuncted statements is false. (Contributed by AV, 6-Sep-2018.) (Proof shortened by AV, 13-Oct-2018.) (Proof shortened by Wolf Lammen, 19-Jan-2020.)

Assertion

Ref	Expression
ornld	⊢ (𝜑 → (((𝜑 → (𝜃 ∨ 𝜏)) ∧ ¬ 𝜃) → 𝜏))

Proof of Theorem ornld

Step	Hyp	Ref	Expression
1		pm3.35 800	. . 3 ⊢ ((𝜑 ∧ (𝜑 → (𝜃 ∨ 𝜏))) → (𝜃 ∨ 𝜏))
2	1	ord 861	. 2 ⊢ ((𝜑 ∧ (𝜑 → (𝜃 ∨ 𝜏))) → (¬ 𝜃 → 𝜏))
3	2	expimpd 454	1 ⊢ (𝜑 → (((𝜑 → (𝜃 ∨ 𝜏)) ∧ ¬ 𝜃) → 𝜏))

Syntax hints:  ¬ wn 3   → wi 4   ∧ wa 396   ∨ wo 844

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8

This theorem depends on definitions:  df-bi 206  df-an 397  df-or 845

This theorem is referenced by:  friendshipgt3  28762  ralralimp  44770