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Theorem nnssi3 33865
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 3949 . . 3 (𝐴 ∈ ℕ → 𝐴𝐷)
31sseli 3949 . . 3 (𝐵 ∈ ℕ → 𝐵𝐷)
41sseli 3949 . . 3 (𝐶 ∈ ℕ → 𝐶𝐷)
52, 3, 43anim123i 1148 . 2 ((𝐴 ∈ ℕ ∧ 𝐵 ∈ ℕ ∧ 𝐶 ∈ ℕ) → (𝐴𝐷𝐵𝐷𝐶𝐷))
6 nnssi3.2 . . 3 (𝐶 ∈ ℕ → 𝜑)
763ad2ant3 1132 . 2 ((𝐴 ∈ ℕ ∧ 𝐵 ∈ ℕ ∧ 𝐶 ∈ ℕ) → 𝜑)
8 nnssi3.3 . 2 (((𝐴𝐷𝐵𝐷𝐶𝐷) ∧ 𝜑) → 𝜓)
95, 7, 8syl2anc 587 1 ((𝐴 ∈ ℕ ∧ 𝐵 ∈ ℕ ∧ 𝐶 ∈ ℕ) → 𝜓)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 399  w3a 1084  wcel 2115  wss 3919  cn 11634
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1912  ax-6 1971  ax-7 2016  ax-8 2117  ax-9 2125  ax-ext 2796
This theorem depends on definitions:  df-bi 210  df-an 400  df-3an 1086  df-ex 1782  df-sb 2071  df-clab 2803  df-cleq 2817  df-clel 2896  df-v 3482  df-in 3926  df-ss 3936
This theorem is referenced by: (None)
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