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Theorem r2ex 2414
Description: Double restricted existential quantification. (Contributed by NM, 11-Nov-1995.)
Assertion
Ref Expression
r2ex (∃𝑥𝐴𝑦𝐵 𝜑 ↔ ∃𝑥𝑦((𝑥𝐴𝑦𝐵) ∧ 𝜑))
Distinct variable groups:   𝑥,𝑦   𝑦,𝐴
Allowed substitution hints:   𝜑(𝑥,𝑦)   𝐴(𝑥)   𝐵(𝑥,𝑦)

Proof of Theorem r2ex
StepHypRef Expression
1 nfcv 2240 . 2 𝑦𝐴
21r2exf 2412 1 (∃𝑥𝐴𝑦𝐵 𝜑 ↔ ∃𝑥𝑦((𝑥𝐴𝑦𝐵) ∧ 𝜑))
Colors of variables: wff set class
Syntax hints:  wa 103  wb 104  wex 1436  wcel 1448  wrex 2376
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-io 671  ax-5 1391  ax-7 1392  ax-gen 1393  ax-ie1 1437  ax-ie2 1438  ax-8 1450  ax-10 1451  ax-11 1452  ax-i12 1453  ax-bndl 1454  ax-4 1455  ax-17 1474  ax-i9 1478  ax-ial 1482  ax-i5r 1483  ax-ext 2082
This theorem depends on definitions:  df-bi 116  df-nf 1405  df-sb 1704  df-cleq 2093  df-clel 2096  df-nfc 2229  df-rex 2381
This theorem is referenced by:  reean  2557  rexiunxp  4619  rnoprab2  5787  genprndl  7230  genprndu  7231  genpdisj  7232  prmuloc  7275  mullocpr  7280  axcnre  7566
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