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Theorem eeanv 2361
Description: Distribute a pair of existential quantifiers over a conjunction. Combination of 19.41v 1941 and 19.42v 1945. For a version requiring fewer axioms but with additional disjoint variable conditions, see exdistrv 1947. (Contributed by NM, 26-Jul-1995.)
Assertion
Ref Expression
eeanv (∃𝑥𝑦(𝜑𝜓) ↔ (∃𝑥𝜑 ∧ ∃𝑦𝜓))
Distinct variable groups:   𝜑,𝑦   𝜓,𝑥
Allowed substitution hints:   𝜑(𝑥)   𝜓(𝑦)

Proof of Theorem eeanv
StepHypRef Expression
1 nfv 1906 . 2 𝑦𝜑
2 nfv 1906 . 2 𝑥𝜓
31, 2eean 2360 1 (∃𝑥𝑦(𝜑𝜓) ↔ (∃𝑥𝜑 ∧ ∃𝑦𝜓))
Colors of variables: wff setvar class
Syntax hints:  wb 207  wa 396  wex 1771
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1787  ax-4 1801  ax-5 1902  ax-6 1961  ax-7 2006  ax-10 2136  ax-11 2151  ax-12 2167
This theorem depends on definitions:  df-bi 208  df-an 397  df-or 842  df-ex 1772  df-nf 1776
This theorem is referenced by:  eeeanv  2362  ee4anv  2363
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