Theorem oranabs

Description: Absorb a disjunct into a conjunct. (Contributed by Roy F. Longton, 23-Jun-2005) (Proof shortened by Wolf Lammen, 10-Nov-2013)

	Ref	Expression
Assertion	oranabs	⊢ ( ( ( 𝜑 ∨ ¬ 𝜓 ) ∧ 𝜓 ) ↔ ( 𝜑 ∧ 𝜓 ) )

Proof

Step	Hyp	Ref	Expression
1		biortn	⊢ ( 𝜓 → ( 𝜑 ↔ ( ¬ 𝜓 ∨ 𝜑 ) ) )
2		orcom	⊢ ( ( ¬ 𝜓 ∨ 𝜑 ) ↔ ( 𝜑 ∨ ¬ 𝜓 ) )
3	1 2	bitr2di	⊢ ( 𝜓 → ( ( 𝜑 ∨ ¬ 𝜓 ) ↔ 𝜑 ) )
4	3	pm5.32ri	⊢ ( ( ( 𝜑 ∨ ¬ 𝜓 ) ∧ 𝜓 ) ↔ ( 𝜑 ∧ 𝜓 ) )