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Theorem rsp2e 2389
 Description: Restricted specialization. (Contributed by FL, 4-Jun-2012.)
Assertion
Ref Expression
rsp2e ((𝑥𝐴𝑦𝐵𝜑) → ∃𝑥𝐴𝑦𝐵 𝜑)

Proof of Theorem rsp2e
StepHypRef Expression
1 simp1 915 . . 3 ((𝑥𝐴𝑦𝐵𝜑) → 𝑥𝐴)
2 rspe 2387 . . . 4 ((𝑦𝐵𝜑) → ∃𝑦𝐵 𝜑)
323adant1 933 . . 3 ((𝑥𝐴𝑦𝐵𝜑) → ∃𝑦𝐵 𝜑)
4 19.8a 1498 . . 3 ((𝑥𝐴 ∧ ∃𝑦𝐵 𝜑) → ∃𝑥(𝑥𝐴 ∧ ∃𝑦𝐵 𝜑))
51, 3, 4syl2anc 397 . 2 ((𝑥𝐴𝑦𝐵𝜑) → ∃𝑥(𝑥𝐴 ∧ ∃𝑦𝐵 𝜑))
6 df-rex 2329 . 2 (∃𝑥𝐴𝑦𝐵 𝜑 ↔ ∃𝑥(𝑥𝐴 ∧ ∃𝑦𝐵 𝜑))
75, 6sylibr 141 1 ((𝑥𝐴𝑦𝐵𝜑) → ∃𝑥𝐴𝑦𝐵 𝜑)
 Colors of variables: wff set class Syntax hints:   → wi 4   ∧ wa 101   ∧ w3a 896  ∃wex 1397   ∈ wcel 1409  ∃wrex 2324 This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 103  ax-ia2 104  ax-ia3 105  ax-gen 1354  ax-ie1 1398  ax-ie2 1399  ax-4 1416 This theorem depends on definitions:  df-bi 114  df-3an 898  df-rex 2329 This theorem is referenced by: (None)
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