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Theorem eeanv 2352
Description: Distribute a pair of existential quantifiers over a conjunction. Combination of 19.41v 1957 and 19.42v 1961. For a version requiring fewer axioms but with additional disjoint variable conditions, see exdistrv 1963. (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 1921 . 2 𝑦𝜑
2 nfv 1921 . 2 𝑥𝜓
31, 2eean 2351 1 (∃𝑥𝑦(𝜑𝜓) ↔ (∃𝑥𝜑 ∧ ∃𝑦𝜓))
Colors of variables: wff setvar class
Syntax hints:  wb 209  wa 399  wex 1786
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1802  ax-4 1816  ax-5 1917  ax-6 1975  ax-7 2020  ax-10 2145  ax-11 2162  ax-12 2179
This theorem depends on definitions:  df-bi 210  df-an 400  df-or 847  df-ex 1787  df-nf 1791
This theorem is referenced by:  eeeanv  2353  ee4anv  2354
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