Theorem nnssi2 37061

Description: Convert a theorem for real/complex numbers into one for positive integers. (Contributed by Jeff Hoffman, 17-Jun-2008.)

Hypotheses

Ref	Expression
nnssi2.1	⊢ ℕ ⊆ 𝐷
nnssi2.2	⊢ (𝐵 ∈ ℕ → 𝜑)
nnssi2.3	⊢ ((𝐴 ∈ 𝐷 ∧ 𝐵 ∈ 𝐷 ∧ 𝜑) → 𝜓)

Assertion

Ref	Expression
nnssi2	⊢ ((𝐴 ∈ ℕ ∧ 𝐵 ∈ ℕ) → 𝜓)

Proof of Theorem nnssi2

Step	Hyp	Ref	Expression
1		nnssi2.1	. . . . 5 ⊢ ℕ ⊆ 𝐷
2	1	sseli 3930	. . . 4 ⊢ (𝐴 ∈ ℕ → 𝐴 ∈ 𝐷)
3	1	sseli 3930	. . . 4 ⊢ (𝐵 ∈ ℕ → 𝐵 ∈ 𝐷)
4		nnssi2.2	. . . 4 ⊢ (𝐵 ∈ ℕ → 𝜑)
5	2, 3, 4	3anim123i 1169	. . 3 ⊢ ((𝐴 ∈ ℕ ∧ 𝐵 ∈ ℕ ∧ 𝐵 ∈ ℕ) → (𝐴 ∈ 𝐷 ∧ 𝐵 ∈ 𝐷 ∧ 𝜑))
6	5	3anidm23 1448	. 2 ⊢ ((𝐴 ∈ ℕ ∧ 𝐵 ∈ ℕ) → (𝐴 ∈ 𝐷 ∧ 𝐵 ∈ 𝐷 ∧ 𝜑))
7		nnssi2.3	. 2 ⊢ ((𝐴 ∈ 𝐷 ∧ 𝐵 ∈ 𝐷 ∧ 𝜑) → 𝜓)
8	6, 7	syl 18	1 ⊢ ((𝐴 ∈ ℕ ∧ 𝐵 ∈ ℕ) → 𝜓)

This proof depends on syntax axioms:   → wi 4   ∧ wa 401   ∧ w3a 1103   ∈ wcel 2145   ⊆ wss 3902  ℕcn 12260

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 2837  df-ss 3919

This theorem is used by:  nndivsub  37063