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Theorem reean 3347
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 nfv 1915 . . . 4 𝑦 𝑥𝐴
2 reean.1 . . . 4 𝑦𝜑
31, 2nfan 1900 . . 3 𝑦(𝑥𝐴𝜑)
4 nfv 1915 . . . 4 𝑥 𝑦𝐵
5 reean.2 . . . 4 𝑥𝜓
64, 5nfan 1900 . . 3 𝑥(𝑦𝐵𝜓)
73, 6eean 2370 . 2 (∃𝑥𝑦((𝑥𝐴𝜑) ∧ (𝑦𝐵𝜓)) ↔ (∃𝑥(𝑥𝐴𝜑) ∧ ∃𝑦(𝑦𝐵𝜓)))
87reeanlem 3346 1 (∃𝑥𝐴𝑦𝐵 (𝜑𝜓) ↔ (∃𝑥𝐴 𝜑 ∧ ∃𝑦𝐵 𝜓))
Colors of variables: wff setvar class
Syntax hints:  wb 209  wa 399  wnf 1785  wcel 2114  wrex 3131
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1911  ax-6 1970  ax-7 2015  ax-10 2145  ax-11 2161  ax-12 2178
This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-tru 1541  df-ex 1782  df-nf 1786  df-ral 3135  df-rex 3136
This theorem is referenced by:  disjrnmpt2  41753
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