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Theorem nnssi3 34741
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 3928 . . 3 (𝐴 ∈ ℕ → 𝐴𝐷)
31sseli 3928 . . 3 (𝐵 ∈ ℕ → 𝐵𝐷)
41sseli 3928 . . 3 (𝐶 ∈ ℕ → 𝐶𝐷)
52, 3, 43anim123i 1150 . 2 ((𝐴 ∈ ℕ ∧ 𝐵 ∈ ℕ ∧ 𝐶 ∈ ℕ) → (𝐴𝐷𝐵𝐷𝐶𝐷))
6 nnssi3.2 . . 3 (𝐶 ∈ ℕ → 𝜑)
763ad2ant3 1134 . 2 ((𝐴 ∈ ℕ ∧ 𝐵 ∈ ℕ ∧ 𝐶 ∈ ℕ) → 𝜑)
8 nnssi3.3 . 2 (((𝐴𝐷𝐵𝐷𝐶𝐷) ∧ 𝜑) → 𝜓)
95, 7, 8syl2anc 584 1 ((𝐴 ∈ ℕ ∧ 𝐵 ∈ ℕ ∧ 𝐶 ∈ ℕ) → 𝜓)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 396  w3a 1086  wcel 2105  wss 3898  cn 12074
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1912  ax-6 1970  ax-7 2010  ax-8 2107  ax-9 2115  ax-ext 2707
This theorem depends on definitions:  df-bi 206  df-an 397  df-3an 1088  df-tru 1543  df-ex 1781  df-sb 2067  df-clab 2714  df-cleq 2728  df-clel 2814  df-v 3443  df-in 3905  df-ss 3915
This theorem is referenced by: (None)
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