Theorem nnssi2 33916

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

Hypotheses

Ref	Expression
nnssi2.1	⊢ ℕ ⊆ 𝐷
nnssi2.2	⊢ (𝐵 ∈ ℕ → 𝜑)
nnssi2.3	⊢ ((𝐴 ∈ 𝐷 ∧ 𝐵 ∈ 𝐷 ∧ 𝜑) → 𝜓)

Assertion

Ref	Expression
nnssi2	⊢ ((𝐴 ∈ ℕ ∧ 𝐵 ∈ ℕ) → 𝜓)

Proof of Theorem nnssi2

Step	Hyp	Ref	Expression
1		nnssi2.1	. . . . 5 ⊢ ℕ ⊆ 𝐷
2	1	sseli 3911	. . . 4 ⊢ (𝐴 ∈ ℕ → 𝐴 ∈ 𝐷)
3	1	sseli 3911	. . . 4 ⊢ (𝐵 ∈ ℕ → 𝐵 ∈ 𝐷)
4		nnssi2.2	. . . 4 ⊢ (𝐵 ∈ ℕ → 𝜑)
5	2, 3, 4	3anim123i 1148	. . 3 ⊢ ((𝐴 ∈ ℕ ∧ 𝐵 ∈ ℕ ∧ 𝐵 ∈ ℕ) → (𝐴 ∈ 𝐷 ∧ 𝐵 ∈ 𝐷 ∧ 𝜑))
6	5	3anidm23 1418	. 2 ⊢ ((𝐴 ∈ ℕ ∧ 𝐵 ∈ ℕ) → (𝐴 ∈ 𝐷 ∧ 𝐵 ∈ 𝐷 ∧ 𝜑))
7		nnssi2.3	. 2 ⊢ ((𝐴 ∈ 𝐷 ∧ 𝐵 ∈ 𝐷 ∧ 𝜑) → 𝜓)
8	6, 7	syl 17	1 ⊢ ((𝐴 ∈ ℕ ∧ 𝐵 ∈ ℕ) → 𝜓)

Syntax hints:   → wi 4   ∧ wa 399   ∧ w3a 1084   ∈ wcel 2111   ⊆ wss 3881  ℕcn 11625

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1911  ax-6 1970  ax-7 2015  ax-8 2113  ax-9 2121  ax-ext 2770

This theorem depends on definitions:  df-bi 210  df-an 400  df-3an 1086  df-ex 1782  df-sb 2070  df-clab 2777  df-cleq 2791  df-clel 2870  df-v 3443  df-in 3888  df-ss 3898

This theorem is referenced by:  nndivsub  33918