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Theorem rsp2 3084
 Description: Restricted specialization, with two quantifiers. (Contributed by NM, 11-Feb-1997.)
Assertion
Ref Expression
rsp2 (∀𝑥𝐴𝑦𝐵 𝜑 → ((𝑥𝐴𝑦𝐵) → 𝜑))

Proof of Theorem rsp2
StepHypRef Expression
1 rsp 3077 . . 3 (∀𝑥𝐴𝑦𝐵 𝜑 → (𝑥𝐴 → ∀𝑦𝐵 𝜑))
2 rsp 3077 . . 3 (∀𝑦𝐵 𝜑 → (𝑦𝐵𝜑))
31, 2syl6 35 . 2 (∀𝑥𝐴𝑦𝐵 𝜑 → (𝑥𝐴 → (𝑦𝐵𝜑)))
43impd 396 1 (∀𝑥𝐴𝑦𝐵 𝜑 → ((𝑥𝐴𝑦𝐵) → 𝜑))
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ∧ wa 382   ∈ wcel 2144  ∀wral 3060 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1869  ax-4 1884  ax-5 1990  ax-6 2056  ax-7 2092  ax-12 2202 This theorem depends on definitions:  df-bi 197  df-an 383  df-ex 1852  df-ral 3065 This theorem is referenced by:  ralcom2  3251  disjxiun  4781  solin  5193  mpt2curryd  7546  cmncom  18415  cnmpt21  21694  cnmpt2t  21696  cnmpt22  21697  cnmptcom  21701  frgrwopreglem5ALT  27501  htthlem  28108  prtlem14  34675  islptre  40363  sprsymrelfolem2  42261
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