Theorem raluz2

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

	Ref	Expression
Assertion	raluz2	⊢ ( ∀ 𝑛 ∈ ( ℤ≥ ‘ 𝑀 ) 𝜑 ↔ ( 𝑀 ∈ ℤ → ∀ 𝑛 ∈ ℤ ( 𝑀 ≤ 𝑛 → 𝜑 ) ) )

Proof

Step	Hyp	Ref	Expression
1		eluz2	⊢ ( 𝑛 ∈ ( ℤ≥ ‘ 𝑀 ) ↔ ( 𝑀 ∈ ℤ ∧ 𝑛 ∈ ℤ ∧ 𝑀 ≤ 𝑛 ) )
2		3anass	⊢ ( ( 𝑀 ∈ ℤ ∧ 𝑛 ∈ ℤ ∧ 𝑀 ≤ 𝑛 ) ↔ ( 𝑀 ∈ ℤ ∧ ( 𝑛 ∈ ℤ ∧ 𝑀 ≤ 𝑛 ) ) )
3	1 2	bitri	⊢ ( 𝑛 ∈ ( ℤ≥ ‘ 𝑀 ) ↔ ( 𝑀 ∈ ℤ ∧ ( 𝑛 ∈ ℤ ∧ 𝑀 ≤ 𝑛 ) ) )
4	3	imbi1i	⊢ ( ( 𝑛 ∈ ( ℤ≥ ‘ 𝑀 ) → 𝜑 ) ↔ ( ( 𝑀 ∈ ℤ ∧ ( 𝑛 ∈ ℤ ∧ 𝑀 ≤ 𝑛 ) ) → 𝜑 ) )
5		impexp	⊢ ( ( ( 𝑀 ∈ ℤ ∧ ( 𝑛 ∈ ℤ ∧ 𝑀 ≤ 𝑛 ) ) → 𝜑 ) ↔ ( 𝑀 ∈ ℤ → ( ( 𝑛 ∈ ℤ ∧ 𝑀 ≤ 𝑛 ) → 𝜑 ) ) )
6		impexp	⊢ ( ( ( 𝑛 ∈ ℤ ∧ 𝑀 ≤ 𝑛 ) → 𝜑 ) ↔ ( 𝑛 ∈ ℤ → ( 𝑀 ≤ 𝑛 → 𝜑 ) ) )
7	6	imbi2i	⊢ ( ( 𝑀 ∈ ℤ → ( ( 𝑛 ∈ ℤ ∧ 𝑀 ≤ 𝑛 ) → 𝜑 ) ) ↔ ( 𝑀 ∈ ℤ → ( 𝑛 ∈ ℤ → ( 𝑀 ≤ 𝑛 → 𝜑 ) ) ) )
8	5 7	bitri	⊢ ( ( ( 𝑀 ∈ ℤ ∧ ( 𝑛 ∈ ℤ ∧ 𝑀 ≤ 𝑛 ) ) → 𝜑 ) ↔ ( 𝑀 ∈ ℤ → ( 𝑛 ∈ ℤ → ( 𝑀 ≤ 𝑛 → 𝜑 ) ) ) )
9		bi2.04	⊢ ( ( 𝑀 ∈ ℤ → ( 𝑛 ∈ ℤ → ( 𝑀 ≤ 𝑛 → 𝜑 ) ) ) ↔ ( 𝑛 ∈ ℤ → ( 𝑀 ∈ ℤ → ( 𝑀 ≤ 𝑛 → 𝜑 ) ) ) )
10	8 9	bitri	⊢ ( ( ( 𝑀 ∈ ℤ ∧ ( 𝑛 ∈ ℤ ∧ 𝑀 ≤ 𝑛 ) ) → 𝜑 ) ↔ ( 𝑛 ∈ ℤ → ( 𝑀 ∈ ℤ → ( 𝑀 ≤ 𝑛 → 𝜑 ) ) ) )
11	4 10	bitri	⊢ ( ( 𝑛 ∈ ( ℤ≥ ‘ 𝑀 ) → 𝜑 ) ↔ ( 𝑛 ∈ ℤ → ( 𝑀 ∈ ℤ → ( 𝑀 ≤ 𝑛 → 𝜑 ) ) ) )
12	11	ralbii2	⊢ ( ∀ 𝑛 ∈ ( ℤ≥ ‘ 𝑀 ) 𝜑 ↔ ∀ 𝑛 ∈ ℤ ( 𝑀 ∈ ℤ → ( 𝑀 ≤ 𝑛 → 𝜑 ) ) )
13		r19.21v	⊢ ( ∀ 𝑛 ∈ ℤ ( 𝑀 ∈ ℤ → ( 𝑀 ≤ 𝑛 → 𝜑 ) ) ↔ ( 𝑀 ∈ ℤ → ∀ 𝑛 ∈ ℤ ( 𝑀 ≤ 𝑛 → 𝜑 ) ) )
14	12 13	bitri	⊢ ( ∀ 𝑛 ∈ ( ℤ≥ ‘ 𝑀 ) 𝜑 ↔ ( 𝑀 ∈ ℤ → ∀ 𝑛 ∈ ℤ ( 𝑀 ≤ 𝑛 → 𝜑 ) ) )