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Theorem nnssi2 32791
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 3748 . . . 4 (𝐴 ∈ ℕ → 𝐴𝐷)
31sseli 3748 . . . 4 (𝐵 ∈ ℕ → 𝐵𝐷)
4 nnssi2.2 . . . 4 (𝐵 ∈ ℕ → 𝜑)
52, 3, 43anim123i 1154 . . 3 ((𝐴 ∈ ℕ ∧ 𝐵 ∈ ℕ ∧ 𝐵 ∈ ℕ) → (𝐴𝐷𝐵𝐷𝜑))
653anidm23 1531 . 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 1071  wcel 2145  wss 3723  cn 11222
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1870  ax-4 1885  ax-5 1991  ax-6 2057  ax-7 2093  ax-9 2154  ax-10 2174  ax-11 2190  ax-12 2203  ax-ext 2751
This theorem depends on definitions:  df-bi 197  df-an 383  df-or 837  df-3an 1073  df-tru 1634  df-ex 1853  df-nf 1858  df-sb 2050  df-clab 2758  df-cleq 2764  df-clel 2767  df-in 3730  df-ss 3737
This theorem is referenced by:  nndivsub  32793
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