Theorem rsp2 3138

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 3131	. . 3 ⊢ (∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐵 𝜑 → (𝑥 ∈ 𝐴 → ∀𝑦 ∈ 𝐵 𝜑))
2		rsp 3131	. . 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 411	1 ⊢ (∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐵 𝜑 → ((𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐵) → 𝜑))

Syntax hints:   → wi 4   ∧ wa 396   ∈ wcel 2106  ∀wral 3064

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1913  ax-6 1971  ax-7 2011  ax-12 2171

This theorem depends on definitions:  df-bi 206  df-an 397  df-ex 1783  df-ral 3069

This theorem is referenced by:  ralcom2  3290  disjxiun  5071  mpocurryd  8085  cmncom  19403  cnmpt21  22822  cnmpt2t  22824  cnmpt22  22825  cnmptcom  22829  frgrwopreglem5ALT  28686  htthlem  29279  qsidomlem2  31629  cplgredgex  33082  prtlem14  36888  islptre  43160  sprsymrelfolem2  44945