Theorem rsp2 3251

Description: Restricted specialization, with two quantifiers. (Contributed by NM, 11-Feb-1997.)

Assertion

Ref	Expression
rsp2	⊢ (∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐵 𝜑 → ((𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐵) → 𝜑))

Proof of Theorem rsp2

Step	Hyp	Ref	Expression
1		rsp 3222	. . 3 ⊢ (∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐵 𝜑 → (𝑥 ∈ 𝐴 → ∀𝑦 ∈ 𝐵 𝜑))
2		rsp 3222	. . 3 ⊢ (∀𝑦 ∈ 𝐵 𝜑 → (𝑦 ∈ 𝐵 → 𝜑))
3	1, 2	syl6 35	. 2 ⊢ (∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐵 𝜑 → (𝑥 ∈ 𝐴 → (𝑦 ∈ 𝐵 → 𝜑)))
4	3	impd 410	1 ⊢ (∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐵 𝜑 → ((𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐵) → 𝜑))

Syntax hints:   → wi 4   ∧ wa 395   ∈ wcel 2113  ∀wral 3049

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 1911  ax-6 1968  ax-7 2009  ax-12 2182

This theorem depends on definitions:  df-bi 207  df-an 396  df-ex 1781  df-ral 3050

This theorem is referenced by:  ralcom2  3345  disjxiun  5093  mpocurryd  8209  cmncom  19725  cnmpt21  23613  cnmpt2t  23615  cnmpt22  23616  cnmptcom  23620  frgrwopreglem5ALT  30346  htthlem  30941  qsidomlem2  33483  cplgredgex  35264  disjlem14  38996  prtlem14  39073  islptre  45807  sprsymrelfolem2  47681