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Theorem nnssi3 31625
 Description: Convert a theorem for real/complex numbers into one for positive integers. (Contributed by Jeff Hoffman, 17-Jun-2008.)
Hypotheses
Ref Expression
nnssi3.1 ℕ ⊆ 𝐷
nnssi3.2 (𝐶 ∈ ℕ → 𝜑)
nnssi3.3 (((𝐴𝐷𝐵𝐷𝐶𝐷) ∧ 𝜑) → 𝜓)
Assertion
Ref Expression
nnssi3 ((𝐴 ∈ ℕ ∧ 𝐵 ∈ ℕ ∧ 𝐶 ∈ ℕ) → 𝜓)

Proof of Theorem nnssi3
StepHypRef Expression
1 nnssi3.1 . . . 4 ℕ ⊆ 𝐷
21sseli 3564 . . 3 (𝐴 ∈ ℕ → 𝐴𝐷)
31sseli 3564 . . 3 (𝐵 ∈ ℕ → 𝐵𝐷)
41sseli 3564 . . 3 (𝐶 ∈ ℕ → 𝐶𝐷)
52, 3, 43anim123i 1240 . 2 ((𝐴 ∈ ℕ ∧ 𝐵 ∈ ℕ ∧ 𝐶 ∈ ℕ) → (𝐴𝐷𝐵𝐷𝐶𝐷))
6 nnssi3.2 . . 3 (𝐶 ∈ ℕ → 𝜑)
763ad2ant3 1077 . 2 ((𝐴 ∈ ℕ ∧ 𝐵 ∈ ℕ ∧ 𝐶 ∈ ℕ) → 𝜑)
8 nnssi3.3 . 2 (((𝐴𝐷𝐵𝐷𝐶𝐷) ∧ 𝜑) → 𝜓)
95, 7, 8syl2anc 691 1 ((𝐴 ∈ ℕ ∧ 𝐵 ∈ ℕ ∧ 𝐶 ∈ ℕ) → 𝜓)
 Colors of variables: wff setvar class 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: (None)
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