Theorem nnssi2 31624

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 3564	. . . 4 ⊢ (𝐴 ∈ ℕ → 𝐴 ∈ 𝐷)
3	1	sseli 3564	. . . 4 ⊢ (𝐵 ∈ ℕ → 𝐵 ∈ 𝐷)
4		nnssi2.2	. . . 4 ⊢ (𝐵 ∈ ℕ → 𝜑)
5	2, 3, 4	3anim123i 1240	. . 3 ⊢ ((𝐴 ∈ ℕ ∧ 𝐵 ∈ ℕ ∧ 𝐵 ∈ ℕ) → (𝐴 ∈ 𝐷 ∧ 𝐵 ∈ 𝐷 ∧ 𝜑))
6	5	3anidm23 1377	. 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 383   ∧ w3a 1031   ∈ wcel 1977   ⊆ wss 3540  ℕcn 10897

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1713  ax-4 1728  ax-5 1827  ax-6 1875  ax-7 1922  ax-10 2006  ax-11 2021  ax-12 2034  ax-13 2234  ax-ext 2590

This theorem depends on definitions:  df-bi 196  df-or 384  df-an 385  df-3an 1033  df-tru 1478  df-ex 1696  df-nf 1701  df-sb 1868  df-clab 2597  df-cleq 2603  df-clel 2606  df-in 3547  df-ss 3554

This theorem is referenced by:  nndivsub  31626