Theorem orfa 36167

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

Step	Hyp	Ref	Expression
1		orcom 866	. . . 4 ⊢ ((𝜑 ∨ ⊥) ↔ (⊥ ∨ 𝜑))
2		df-or 844	. . . 4 ⊢ ((⊥ ∨ 𝜑) ↔ (¬ ⊥ → 𝜑))
3	1, 2	bitri 274	. . 3 ⊢ ((𝜑 ∨ ⊥) ↔ (¬ ⊥ → 𝜑))
4		fal 1553	. . . 4 ⊢ ¬ ⊥
5		pm2.27 42	. . . 4 ⊢ (¬ ⊥ → ((¬ ⊥ → 𝜑) → 𝜑))
6	4, 5	ax-mp 5	. . 3 ⊢ ((¬ ⊥ → 𝜑) → 𝜑)
7	3, 6	sylbi 216	. 2 ⊢ ((𝜑 ∨ ⊥) → 𝜑)
8		orc 863	. 2 ⊢ (𝜑 → (𝜑 ∨ ⊥))
9	7, 8	impbii 208	1 ⊢ ((𝜑 ∨ ⊥) ↔ 𝜑)

Syntax hints:  ¬ wn 3   → wi 4   ↔ wb 205   ∨ wo 843  ⊥wfal 1551

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-or 844  df-tru 1542  df-fal 1552

This theorem is referenced by: (None)