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Theorem rsp2e 2486
Description: Restricted specialization. (Contributed by FL, 4-Jun-2012.)
Assertion
Ref Expression
rsp2e ((𝑥𝐴𝑦𝐵𝜑) → ∃𝑥𝐴𝑦𝐵 𝜑)

Proof of Theorem rsp2e
StepHypRef Expression
1 simp1 982 . . 3 ((𝑥𝐴𝑦𝐵𝜑) → 𝑥𝐴)
2 rspe 2484 . . . 4 ((𝑦𝐵𝜑) → ∃𝑦𝐵 𝜑)
323adant1 1000 . . 3 ((𝑥𝐴𝑦𝐵𝜑) → ∃𝑦𝐵 𝜑)
4 19.8a 1570 . . 3 ((𝑥𝐴 ∧ ∃𝑦𝐵 𝜑) → ∃𝑥(𝑥𝐴 ∧ ∃𝑦𝐵 𝜑))
51, 3, 4syl2anc 409 . 2 ((𝑥𝐴𝑦𝐵𝜑) → ∃𝑥(𝑥𝐴 ∧ ∃𝑦𝐵 𝜑))
6 df-rex 2423 . 2 (∃𝑥𝐴𝑦𝐵 𝜑 ↔ ∃𝑥(𝑥𝐴 ∧ ∃𝑦𝐵 𝜑))
75, 6sylibr 133 1 ((𝑥𝐴𝑦𝐵𝜑) → ∃𝑥𝐴𝑦𝐵 𝜑)
Colors of variables: wff set class
Syntax hints:  wi 4  wa 103  w3a 963  wex 1469  wcel 1481  wrex 2418
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-gen 1426  ax-ie1 1470  ax-ie2 1471  ax-4 1488
This theorem depends on definitions:  df-bi 116  df-3an 965  df-rex 2423
This theorem is referenced by: (None)
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