MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  rsp2 Structured version   Visualization version   GIF version

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
  Copyright terms: Public domain W3C validator