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Theorem orfa 32854
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 400 . . . 4 ((𝜑 ∨ ⊥) ↔ (⊥ ∨ 𝜑))
2 df-or 383 . . . 4 ((⊥ ∨ 𝜑) ↔ (¬ ⊥ → 𝜑))
31, 2bitri 262 . . 3 ((𝜑 ∨ ⊥) ↔ (¬ ⊥ → 𝜑))
4 fal 1481 . . . 4 ¬ ⊥
5 pm2.27 40 . . . 4 (¬ ⊥ → ((¬ ⊥ → 𝜑) → 𝜑))
64, 5ax-mp 5 . . 3 ((¬ ⊥ → 𝜑) → 𝜑)
73, 6sylbi 205 . 2 ((𝜑 ∨ ⊥) → 𝜑)
8 orc 398 . 2 (𝜑 → (𝜑 ∨ ⊥))
97, 8impbii 197 1 ((𝜑 ∨ ⊥) ↔ 𝜑)
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wb 194  wo 381  wfal 1479
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 195  df-or 383  df-tru 1477  df-fal 1480
This theorem is referenced by: (None)
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