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Theorem rsp2 2935
 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 2928 . . 3 (∀𝑥𝐴𝑦𝐵 𝜑 → (𝑥𝐴 → ∀𝑦𝐵 𝜑))
2 rsp 2928 . . 3 (∀𝑦𝐵 𝜑 → (𝑦𝐵𝜑))
31, 2syl6 35 . 2 (∀𝑥𝐴𝑦𝐵 𝜑 → (𝑥𝐴 → (𝑦𝐵𝜑)))
43impd 447 1 (∀𝑥𝐴𝑦𝐵 𝜑 → ((𝑥𝐴𝑦𝐵) → 𝜑))
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ∧ wa 384   ∈ wcel 1989  ∀wral 2911 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1721  ax-4 1736  ax-5 1838  ax-6 1887  ax-7 1934  ax-12 2046 This theorem depends on definitions:  df-bi 197  df-an 386  df-ex 1704  df-ral 2916 This theorem is referenced by:  ralcom2  3102  disjxiun  4647  disjxiunOLD  4648  solin  5056  mpt2curryd  7392  cmncom  18203  cnmpt21  21468  cnmpt2t  21470  cnmpt22  21471  cnmptcom  21475  frgrwopreglem5  27171  htthlem  27758  prtlem14  33985  islptre  39657  sprsymrelfolem2  41514
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