Metamath Proof Explorer


Theorem nnssi3

Description: Convert a theorem for real/complex numbers into one for positive integers. (Contributed by Jeff Hoffman, 17-Jun-2008)

Ref Expression
Hypotheses nnssi3.1 ℕ ⊆ 𝐷
nnssi3.2 ( 𝐶 ∈ ℕ → 𝜑 )
nnssi3.3 ( ( ( 𝐴𝐷𝐵𝐷𝐶𝐷 ) ∧ 𝜑 ) → 𝜓 )
Assertion nnssi3 ( ( 𝐴 ∈ ℕ ∧ 𝐵 ∈ ℕ ∧ 𝐶 ∈ ℕ ) → 𝜓 )

Proof

Step Hyp Ref Expression
1 nnssi3.1 ℕ ⊆ 𝐷
2 nnssi3.2 ( 𝐶 ∈ ℕ → 𝜑 )
3 nnssi3.3 ( ( ( 𝐴𝐷𝐵𝐷𝐶𝐷 ) ∧ 𝜑 ) → 𝜓 )
4 1 sseli ( 𝐴 ∈ ℕ → 𝐴𝐷 )
5 1 sseli ( 𝐵 ∈ ℕ → 𝐵𝐷 )
6 1 sseli ( 𝐶 ∈ ℕ → 𝐶𝐷 )
7 4 5 6 3anim123i ( ( 𝐴 ∈ ℕ ∧ 𝐵 ∈ ℕ ∧ 𝐶 ∈ ℕ ) → ( 𝐴𝐷𝐵𝐷𝐶𝐷 ) )
8 2 3ad2ant3 ( ( 𝐴 ∈ ℕ ∧ 𝐵 ∈ ℕ ∧ 𝐶 ∈ ℕ ) → 𝜑 )
9 7 8 3 syl2anc ( ( 𝐴 ∈ ℕ ∧ 𝐵 ∈ ℕ ∧ 𝐶 ∈ ℕ ) → 𝜓 )