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Theorem orfa 35513
 Description: The falsum ⊥ can be removed from a disjunction. (Contributed by Giovanni Mascellani, 15-Sep-2017.)
Assertion
Ref Expression
orfa ((𝜑 ∨ ⊥) ↔ 𝜑)

Proof of Theorem orfa
StepHypRef Expression
1 orcom 867 . . . 4 ((𝜑 ∨ ⊥) ↔ (⊥ ∨ 𝜑))
2 df-or 845 . . . 4 ((⊥ ∨ 𝜑) ↔ (¬ ⊥ → 𝜑))
31, 2bitri 278 . . 3 ((𝜑 ∨ ⊥) ↔ (¬ ⊥ → 𝜑))
4 fal 1552 . . . 4 ¬ ⊥
5 pm2.27 42 . . . 4 (¬ ⊥ → ((¬ ⊥ → 𝜑) → 𝜑))
64, 5ax-mp 5 . . 3 ((¬ ⊥ → 𝜑) → 𝜑)
73, 6sylbi 220 . 2 ((𝜑 ∨ ⊥) → 𝜑)
8 orc 864 . 2 (𝜑 → (𝜑 ∨ ⊥))
97, 8impbii 212 1 ((𝜑 ∨ ⊥) ↔ 𝜑)
 Colors of variables: wff setvar class Syntax hints:  ¬ wn 3   → wi 4   ↔ wb 209   ∨ wo 844  ⊥wfal 1550 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 210  df-or 845  df-tru 1541  df-fal 1551 This theorem is referenced by: (None)
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