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Theorem r2exf 2390
Description: Double restricted existential quantification. (Contributed by Mario Carneiro, 14-Oct-2016.)
Hypothesis
Ref Expression
r2alf.1 𝑦𝐴
Assertion
Ref Expression
r2exf (∃𝑥𝐴𝑦𝐵 𝜑 ↔ ∃𝑥𝑦((𝑥𝐴𝑦𝐵) ∧ 𝜑))
Distinct variable group:   𝑥,𝑦
Allowed substitution hints:   𝜑(𝑥,𝑦)   𝐴(𝑥,𝑦)   𝐵(𝑥,𝑦)

Proof of Theorem r2exf
StepHypRef Expression
1 df-rex 2359 . 2 (∃𝑥𝐴𝑦𝐵 𝜑 ↔ ∃𝑥(𝑥𝐴 ∧ ∃𝑦𝐵 𝜑))
2 r2alf.1 . . . . . 6 𝑦𝐴
32nfcri 2217 . . . . 5 𝑦 𝑥𝐴
4319.42 1619 . . . 4 (∃𝑦(𝑥𝐴 ∧ (𝑦𝐵𝜑)) ↔ (𝑥𝐴 ∧ ∃𝑦(𝑦𝐵𝜑)))
5 anass 393 . . . . 5 (((𝑥𝐴𝑦𝐵) ∧ 𝜑) ↔ (𝑥𝐴 ∧ (𝑦𝐵𝜑)))
65exbii 1537 . . . 4 (∃𝑦((𝑥𝐴𝑦𝐵) ∧ 𝜑) ↔ ∃𝑦(𝑥𝐴 ∧ (𝑦𝐵𝜑)))
7 df-rex 2359 . . . . 5 (∃𝑦𝐵 𝜑 ↔ ∃𝑦(𝑦𝐵𝜑))
87anbi2i 445 . . . 4 ((𝑥𝐴 ∧ ∃𝑦𝐵 𝜑) ↔ (𝑥𝐴 ∧ ∃𝑦(𝑦𝐵𝜑)))
94, 6, 83bitr4i 210 . . 3 (∃𝑦((𝑥𝐴𝑦𝐵) ∧ 𝜑) ↔ (𝑥𝐴 ∧ ∃𝑦𝐵 𝜑))
109exbii 1537 . 2 (∃𝑥𝑦((𝑥𝐴𝑦𝐵) ∧ 𝜑) ↔ ∃𝑥(𝑥𝐴 ∧ ∃𝑦𝐵 𝜑))
111, 10bitr4i 185 1 (∃𝑥𝐴𝑦𝐵 𝜑 ↔ ∃𝑥𝑦((𝑥𝐴𝑦𝐵) ∧ 𝜑))
Colors of variables: wff set class
Syntax hints:  wa 102  wb 103  wex 1422  wcel 1434  wnfc 2210  wrex 2354
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106  ax-io 663  ax-5 1377  ax-7 1378  ax-gen 1379  ax-ie1 1423  ax-ie2 1424  ax-8 1436  ax-10 1437  ax-11 1438  ax-i12 1439  ax-bndl 1440  ax-4 1441  ax-17 1460  ax-i9 1464  ax-ial 1468  ax-i5r 1469  ax-ext 2065
This theorem depends on definitions:  df-bi 115  df-nf 1391  df-sb 1688  df-cleq 2076  df-clel 2079  df-nfc 2212  df-rex 2359
This theorem is referenced by:  r2ex  2392  rexcomf  2522
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