MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  ornld Structured version   Visualization version   GIF version

Theorem ornld 1053
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 → (𝜃𝜏)) ∧ ¬ 𝜃) → 𝜏))

Proof of Theorem ornld
StepHypRef Expression
1 pm3.35 799 . . 3 ((𝜑 ∧ (𝜑 → (𝜃𝜏))) → (𝜃𝜏))
21ord 858 . 2 ((𝜑 ∧ (𝜑 → (𝜃𝜏))) → (¬ 𝜃𝜏))
32expimpd 454 1 (𝜑 → (((𝜑 → (𝜃𝜏)) ∧ ¬ 𝜃) → 𝜏))
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wa 396  wo 841
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 208  df-an 397  df-or 842
This theorem is referenced by:  friendshipgt3  28104  ralralimp  43354
  Copyright terms: Public domain W3C validator