Theorem nnssi2 35972

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 3978	. . . 4 ⊢ (𝐴 ∈ ℕ → 𝐴 ∈ 𝐷)
3	1	sseli 3978	. . . 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 394   ∧ w3a 1084   ∈ wcel 2098   ⊆ wss 3949  ℕcn 12250

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1905  ax-6 1963  ax-7 2003  ax-8 2100  ax-9 2108  ax-ext 2699

This theorem depends on definitions:  df-bi 206  df-an 395  df-3an 1086  df-tru 1536  df-ex 1774  df-sb 2060  df-clab 2706  df-cleq 2720  df-clel 2806  df-v 3475  df-in 3956  df-ss 3966

This theorem is referenced by:  nndivsub  35974