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Theorem reean 3100
 Description: Rearrange restricted existential quantifiers. (Contributed by NM, 27-Oct-2010.) (Proof shortened by Andrew Salmon, 30-May-2011.)
Hypotheses
Ref Expression
reean.1 𝑦𝜑
reean.2 𝑥𝜓
Assertion
Ref Expression
reean (∃𝑥𝐴𝑦𝐵 (𝜑𝜓) ↔ (∃𝑥𝐴 𝜑 ∧ ∃𝑦𝐵 𝜓))
Distinct variable groups:   𝑦,𝐴   𝑥,𝐵   𝑥,𝑦
Allowed substitution hints:   𝜑(𝑥,𝑦)   𝜓(𝑥,𝑦)   𝐴(𝑥)   𝐵(𝑦)

Proof of Theorem reean
StepHypRef Expression
1 an4 864 . . . 4 (((𝑥𝐴𝑦𝐵) ∧ (𝜑𝜓)) ↔ ((𝑥𝐴𝜑) ∧ (𝑦𝐵𝜓)))
212exbii 1772 . . 3 (∃𝑥𝑦((𝑥𝐴𝑦𝐵) ∧ (𝜑𝜓)) ↔ ∃𝑥𝑦((𝑥𝐴𝜑) ∧ (𝑦𝐵𝜓)))
3 nfv 1840 . . . . 5 𝑦 𝑥𝐴
4 reean.1 . . . . 5 𝑦𝜑
53, 4nfan 1825 . . . 4 𝑦(𝑥𝐴𝜑)
6 nfv 1840 . . . . 5 𝑥 𝑦𝐵
7 reean.2 . . . . 5 𝑥𝜓
86, 7nfan 1825 . . . 4 𝑥(𝑦𝐵𝜓)
95, 8eean 2180 . . 3 (∃𝑥𝑦((𝑥𝐴𝜑) ∧ (𝑦𝐵𝜓)) ↔ (∃𝑥(𝑥𝐴𝜑) ∧ ∃𝑦(𝑦𝐵𝜓)))
102, 9bitri 264 . 2 (∃𝑥𝑦((𝑥𝐴𝑦𝐵) ∧ (𝜑𝜓)) ↔ (∃𝑥(𝑥𝐴𝜑) ∧ ∃𝑦(𝑦𝐵𝜓)))
11 r2ex 3056 . 2 (∃𝑥𝐴𝑦𝐵 (𝜑𝜓) ↔ ∃𝑥𝑦((𝑥𝐴𝑦𝐵) ∧ (𝜑𝜓)))
12 df-rex 2914 . . 3 (∃𝑥𝐴 𝜑 ↔ ∃𝑥(𝑥𝐴𝜑))
13 df-rex 2914 . . 3 (∃𝑦𝐵 𝜓 ↔ ∃𝑦(𝑦𝐵𝜓))
1412, 13anbi12i 732 . 2 ((∃𝑥𝐴 𝜑 ∧ ∃𝑦𝐵 𝜓) ↔ (∃𝑥(𝑥𝐴𝜑) ∧ ∃𝑦(𝑦𝐵𝜓)))
1510, 11, 143bitr4i 292 1 (∃𝑥𝐴𝑦𝐵 (𝜑𝜓) ↔ (∃𝑥𝐴 𝜑 ∧ ∃𝑦𝐵 𝜓))
 Colors of variables: wff setvar class Syntax hints:   ↔ wb 196   ∧ wa 384  ∃wex 1701  Ⅎwnf 1705   ∈ wcel 1987  ∃wrex 2909 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1719  ax-4 1734  ax-5 1836  ax-6 1885  ax-7 1932  ax-10 2016  ax-11 2031  ax-12 2044 This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-tru 1483  df-ex 1702  df-nf 1707  df-ral 2913  df-rex 2914 This theorem is referenced by:  reeanv  3101  disjrnmpt2  38884
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