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Theorem nnssi2 31469
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 ((𝐴𝐷𝐵𝐷𝜑) → 𝜓)
Assertion
Ref Expression
nnssi2 ((𝐴 ∈ ℕ ∧ 𝐵 ∈ ℕ) → 𝜓)

Proof of Theorem nnssi2
StepHypRef Expression
1 nnssi2.1 . . . . 5 ℕ ⊆ 𝐷
21sseli 3468 . . . 4 (𝐴 ∈ ℕ → 𝐴𝐷)
31sseli 3468 . . . 4 (𝐵 ∈ ℕ → 𝐵𝐷)
4 nnssi2.2 . . . 4 (𝐵 ∈ ℕ → 𝜑)
52, 3, 43anim123i 1239 . . 3 ((𝐴 ∈ ℕ ∧ 𝐵 ∈ ℕ ∧ 𝐵 ∈ ℕ) → (𝐴𝐷𝐵𝐷𝜑))
653anidm23 1376 . 2 ((𝐴 ∈ ℕ ∧ 𝐵 ∈ ℕ) → (𝐴𝐷𝐵𝐷𝜑))
7 nnssi2.3 . 2 ((𝐴𝐷𝐵𝐷𝜑) → 𝜓)
86, 7syl 17 1 ((𝐴 ∈ ℕ ∧ 𝐵 ∈ ℕ) → 𝜓)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 382  w3a 1030  wcel 1938  wss 3444  cn 10773
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1700  ax-4 1713  ax-5 1793  ax-6 1838  ax-7 1885  ax-10 1966  ax-11 1971  ax-12 1983  ax-13 2137  ax-ext 2494
This theorem depends on definitions:  df-bi 195  df-an 384  df-3an 1032  df-tru 1477  df-ex 1695  df-nf 1699  df-sb 1831  df-clab 2501  df-cleq 2507  df-clel 2510  df-in 3451  df-ss 3458
This theorem is referenced by:  nndivsub  31471
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