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Theorem rsp2e 3267
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 3238 . . 3 ((𝑦𝐵𝜑) → ∃𝑦𝐵 𝜑)
2 rspe 3238 . . 3 ((𝑥𝐴 ∧ ∃𝑦𝐵 𝜑) → ∃𝑥𝐴𝑦𝐵 𝜑)
31, 2sylan2 592 . 2 ((𝑥𝐴 ∧ (𝑦𝐵𝜑)) → ∃𝑥𝐴𝑦𝐵 𝜑)
433impb 1112 1 ((𝑥𝐴𝑦𝐵𝜑) → ∃𝑥𝐴𝑦𝐵 𝜑)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 395  w3a 1084  wcel 2098  wrex 3062
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1905  ax-6 1963  ax-7 2003  ax-12 2163
This theorem depends on definitions:  df-bi 206  df-an 396  df-3an 1086  df-ex 1774  df-rex 3063
This theorem is referenced by:  pell14qrdich  42121
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