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Theorem nnssi3 32150
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 3584 . . 3 (𝐴 ∈ ℕ → 𝐴𝐷)
31sseli 3584 . . 3 (𝐵 ∈ ℕ → 𝐵𝐷)
41sseli 3584 . . 3 (𝐶 ∈ ℕ → 𝐶𝐷)
52, 3, 43anim123i 1245 . 2 ((𝐴 ∈ ℕ ∧ 𝐵 ∈ ℕ ∧ 𝐶 ∈ ℕ) → (𝐴𝐷𝐵𝐷𝐶𝐷))
6 nnssi3.2 . . 3 (𝐶 ∈ ℕ → 𝜑)
763ad2ant3 1082 . 2 ((𝐴 ∈ ℕ ∧ 𝐵 ∈ ℕ ∧ 𝐶 ∈ ℕ) → 𝜑)
8 nnssi3.3 . 2 (((𝐴𝐷𝐵𝐷𝐶𝐷) ∧ 𝜑) → 𝜓)
95, 7, 8syl2anc 692 1 ((𝐴 ∈ ℕ ∧ 𝐵 ∈ ℕ ∧ 𝐶 ∈ ℕ) → 𝜓)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 384  w3a 1036  wcel 1987  wss 3560  cn 10980
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1719  ax-4 1734  ax-5 1836  ax-6 1885  ax-7 1932  ax-9 1996  ax-10 2016  ax-11 2031  ax-12 2044  ax-13 2245  ax-ext 2601
This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3an 1038  df-tru 1483  df-ex 1702  df-nf 1707  df-sb 1878  df-clab 2608  df-cleq 2614  df-clel 2617  df-in 3567  df-ss 3574
This theorem is referenced by: (None)
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