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Theorem rsp2e 2528
Description: Restricted specialization. (Contributed by FL, 4-Jun-2012.)
Assertion
Ref Expression
rsp2e ((𝑥𝐴𝑦𝐵𝜑) → ∃𝑥𝐴𝑦𝐵 𝜑)

Proof of Theorem rsp2e
StepHypRef Expression
1 simp1 997 . . 3 ((𝑥𝐴𝑦𝐵𝜑) → 𝑥𝐴)
2 rspe 2526 . . . 4 ((𝑦𝐵𝜑) → ∃𝑦𝐵 𝜑)
323adant1 1015 . . 3 ((𝑥𝐴𝑦𝐵𝜑) → ∃𝑦𝐵 𝜑)
4 19.8a 1590 . . 3 ((𝑥𝐴 ∧ ∃𝑦𝐵 𝜑) → ∃𝑥(𝑥𝐴 ∧ ∃𝑦𝐵 𝜑))
51, 3, 4syl2anc 411 . 2 ((𝑥𝐴𝑦𝐵𝜑) → ∃𝑥(𝑥𝐴 ∧ ∃𝑦𝐵 𝜑))
6 df-rex 2461 . 2 (∃𝑥𝐴𝑦𝐵 𝜑 ↔ ∃𝑥(𝑥𝐴 ∧ ∃𝑦𝐵 𝜑))
75, 6sylibr 134 1 ((𝑥𝐴𝑦𝐵𝜑) → ∃𝑥𝐴𝑦𝐵 𝜑)
Colors of variables: wff set class
Syntax hints:  wi 4  wa 104  w3a 978  wex 1492  wcel 2148  wrex 2456
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-gen 1449  ax-ie1 1493  ax-ie2 1494  ax-4 1510
This theorem depends on definitions:  df-bi 117  df-3an 980  df-rex 2461
This theorem is referenced by: (None)
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