Theorem nnssi3 36754

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	⊢ (((𝐴 ∈ 𝐷 ∧ 𝐵 ∈ 𝐷 ∧ 𝐶 ∈ 𝐷) ∧ 𝜑) → 𝜓)

Assertion

Ref	Expression
nnssi3	⊢ ((𝐴 ∈ ℕ ∧ 𝐵 ∈ ℕ ∧ 𝐶 ∈ ℕ) → 𝜓)

Proof of Theorem nnssi3

Step	Hyp	Ref	Expression
1		nnssi3.1	. . . 4 ⊢ ℕ ⊆ 𝐷
2	1	sseli 3923	. . 3 ⊢ (𝐴 ∈ ℕ → 𝐴 ∈ 𝐷)
3	1	sseli 3923	. . 3 ⊢ (𝐵 ∈ ℕ → 𝐵 ∈ 𝐷)
4	1	sseli 3923	. . 3 ⊢ (𝐶 ∈ ℕ → 𝐶 ∈ 𝐷)
5	2, 3, 4	3anim123i 1160	. 2 ⊢ ((𝐴 ∈ ℕ ∧ 𝐵 ∈ ℕ ∧ 𝐶 ∈ ℕ) → (𝐴 ∈ 𝐷 ∧ 𝐵 ∈ 𝐷 ∧ 𝐶 ∈ 𝐷))
6		nnssi3.2	. . 3 ⊢ (𝐶 ∈ ℕ → 𝜑)
7	6	3ad2ant3 1144	. 2 ⊢ ((𝐴 ∈ ℕ ∧ 𝐵 ∈ ℕ ∧ 𝐶 ∈ ℕ) → 𝜑)
8		nnssi3.3	. 2 ⊢ (((𝐴 ∈ 𝐷 ∧ 𝐵 ∈ 𝐷 ∧ 𝐶 ∈ 𝐷) ∧ 𝜑) → 𝜓)
9	5, 7, 8	syl2anc 592	1 ⊢ ((𝐴 ∈ ℕ ∧ 𝐵 ∈ ℕ ∧ 𝐶 ∈ ℕ) → 𝜓)

Syntax hints:   → wi 4   ∧ wa 398   ∧ w3a 1095   ∈ wcel 2132   ⊆ wss 3895  ℕcn 12196

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1805  ax-4 1819  ax-5 1920  ax-6 1977  ax-7 2018  ax-8 2134

This theorem depends on definitions:  df-bi 209  df-an 399  df-3an 1097  df-ex 1790  df-clel 2827  df-ss 3912

This theorem is referenced by: (None)