Theorem raluz

Description: Restricted universal quantification in an upper set of integers. (Contributed by NM, 9-Sep-2005)

	Ref	Expression
Assertion	raluz	⊢ ( 𝑀 ∈ ℤ → ( ∀ 𝑛 ∈ ( ℤ≥ ‘ 𝑀 ) 𝜑 ↔ ∀ 𝑛 ∈ ℤ ( 𝑀 ≤ 𝑛 → 𝜑 ) ) )

Proof

Step	Hyp	Ref	Expression
1		eluz1	⊢ ( 𝑀 ∈ ℤ → ( 𝑛 ∈ ( ℤ≥ ‘ 𝑀 ) ↔ ( 𝑛 ∈ ℤ ∧ 𝑀 ≤ 𝑛 ) ) )
2	1	imbi1d	⊢ ( 𝑀 ∈ ℤ → ( ( 𝑛 ∈ ( ℤ≥ ‘ 𝑀 ) → 𝜑 ) ↔ ( ( 𝑛 ∈ ℤ ∧ 𝑀 ≤ 𝑛 ) → 𝜑 ) ) )
3		impexp	⊢ ( ( ( 𝑛 ∈ ℤ ∧ 𝑀 ≤ 𝑛 ) → 𝜑 ) ↔ ( 𝑛 ∈ ℤ → ( 𝑀 ≤ 𝑛 → 𝜑 ) ) )
4	2 3	syl6bb	⊢ ( 𝑀 ∈ ℤ → ( ( 𝑛 ∈ ( ℤ≥ ‘ 𝑀 ) → 𝜑 ) ↔ ( 𝑛 ∈ ℤ → ( 𝑀 ≤ 𝑛 → 𝜑 ) ) ) )
5	4	ralbidv2	⊢ ( 𝑀 ∈ ℤ → ( ∀ 𝑛 ∈ ( ℤ≥ ‘ 𝑀 ) 𝜑 ↔ ∀ 𝑛 ∈ ℤ ( 𝑀 ≤ 𝑛 → 𝜑 ) ) )