Theorem nnssi3 37166

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 3926	. . 3 ⊢ (𝐴 ∈ ℕ → 𝐴 ∈ 𝐷)
3	1	sseli 3926	. . 3 ⊢ (𝐵 ∈ ℕ → 𝐵 ∈ 𝐷)
4	1	sseli 3926	. . 3 ⊢ (𝐶 ∈ ℕ → 𝐶 ∈ 𝐷)
5	2, 3, 4	3anim123i 1169	. 2 ⊢ ((𝐴 ∈ ℕ ∧ 𝐵 ∈ ℕ ∧ 𝐶 ∈ ℕ) → (𝐴 ∈ 𝐷 ∧ 𝐵 ∈ 𝐷 ∧ 𝐶 ∈ 𝐷))
6		nnssi3.2	. . 3 ⊢ (𝐶 ∈ ℕ → 𝜑)
7	6	3ad2ant3 1153	. 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 596	1 ⊢ ((𝐴 ∈ ℕ ∧ 𝐵 ∈ ℕ ∧ 𝐶 ∈ ℕ) → 𝜓)

This proof depends on syntax axioms:   → wi 4   ∧ wa 401   ∧ w3a 1103   ∈ wcel 2145   ⊆ wss 3898  ℕcn 12304

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1828  ax-4 1842  ax-5 1943  ax-6 2000  ax-7 2041  ax-8 2147

This proof depends on definitions:  df-bi 210  df-an 402  df-3an 1105  df-ex 1813  df-clel 2835  df-ss 3915

This theorem is used by: (None)