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Theorem r2exf 2430
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 2399 . 2 (∃𝑥𝐴𝑦𝐵 𝜑 ↔ ∃𝑥(𝑥𝐴 ∧ ∃𝑦𝐵 𝜑))
2 r2alf.1 . . . . . 6 𝑦𝐴
32nfcri 2252 . . . . 5 𝑦 𝑥𝐴
4319.42 1651 . . . 4 (∃𝑦(𝑥𝐴 ∧ (𝑦𝐵𝜑)) ↔ (𝑥𝐴 ∧ ∃𝑦(𝑦𝐵𝜑)))
5 anass 398 . . . . 5 (((𝑥𝐴𝑦𝐵) ∧ 𝜑) ↔ (𝑥𝐴 ∧ (𝑦𝐵𝜑)))
65exbii 1569 . . . 4 (∃𝑦((𝑥𝐴𝑦𝐵) ∧ 𝜑) ↔ ∃𝑦(𝑥𝐴 ∧ (𝑦𝐵𝜑)))
7 df-rex 2399 . . . . 5 (∃𝑦𝐵 𝜑 ↔ ∃𝑦(𝑦𝐵𝜑))
87anbi2i 452 . . . 4 ((𝑥𝐴 ∧ ∃𝑦𝐵 𝜑) ↔ (𝑥𝐴 ∧ ∃𝑦(𝑦𝐵𝜑)))
94, 6, 83bitr4i 211 . . 3 (∃𝑦((𝑥𝐴𝑦𝐵) ∧ 𝜑) ↔ (𝑥𝐴 ∧ ∃𝑦𝐵 𝜑))
109exbii 1569 . 2 (∃𝑥𝑦((𝑥𝐴𝑦𝐵) ∧ 𝜑) ↔ ∃𝑥(𝑥𝐴 ∧ ∃𝑦𝐵 𝜑))
111, 10bitr4i 186 1 (∃𝑥𝐴𝑦𝐵 𝜑 ↔ ∃𝑥𝑦((𝑥𝐴𝑦𝐵) ∧ 𝜑))
Colors of variables: wff set class
Syntax hints:  wa 103  wb 104  wex 1453  wcel 1465  wnfc 2245  wrex 2394
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-io 683  ax-5 1408  ax-7 1409  ax-gen 1410  ax-ie1 1454  ax-ie2 1455  ax-8 1467  ax-10 1468  ax-11 1469  ax-i12 1470  ax-bndl 1471  ax-4 1472  ax-17 1491  ax-i9 1495  ax-ial 1499  ax-i5r 1500  ax-ext 2099
This theorem depends on definitions:  df-bi 116  df-nf 1422  df-sb 1721  df-cleq 2110  df-clel 2113  df-nfc 2247  df-rex 2399
This theorem is referenced by:  r2ex  2432  rexcomf  2570
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