MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  rsp2e Structured version   Visualization version   GIF version

Theorem rsp2e 3003
Description: Restricted specialization. (Contributed by FL, 4-Jun-2012.) (Proof shortened by Wolf Lammen, 7-Jan-2020.)
Assertion
Ref Expression
rsp2e ((𝑥𝐴𝑦𝐵𝜑) → ∃𝑥𝐴𝑦𝐵 𝜑)

Proof of Theorem rsp2e
StepHypRef Expression
1 rspe 3002 . . 3 ((𝑦𝐵𝜑) → ∃𝑦𝐵 𝜑)
2 rspe 3002 . . 3 ((𝑥𝐴 ∧ ∃𝑦𝐵 𝜑) → ∃𝑥𝐴𝑦𝐵 𝜑)
31, 2sylan2 491 . 2 ((𝑥𝐴 ∧ (𝑦𝐵𝜑)) → ∃𝑥𝐴𝑦𝐵 𝜑)
433impb 1259 1 ((𝑥𝐴𝑦𝐵𝜑) → ∃𝑥𝐴𝑦𝐵 𝜑)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 384  w3a 1037  wcel 1989  wrex 2912
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-3an 1039  df-ex 1704  df-rex 2917
This theorem is referenced by:  pell14qrdich  37259
  Copyright terms: Public domain W3C validator