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Theorem exdistr 1955
 Description: Distribution of existential quantifiers. See also exdistrv 1956. (Contributed by NM, 9-Mar-1995.)
Assertion
Ref Expression
exdistr (∃𝑥𝑦(𝜑𝜓) ↔ ∃𝑥(𝜑 ∧ ∃𝑦𝜓))
Distinct variable group:   𝜑,𝑦
Allowed substitution hints:   𝜑(𝑥)   𝜓(𝑥,𝑦)

Proof of Theorem exdistr
StepHypRef Expression
1 19.42v 1954 . 2 (∃𝑦(𝜑𝜓) ↔ (𝜑 ∧ ∃𝑦𝜓))
21exbii 1849 1 (∃𝑥𝑦(𝜑𝜓) ↔ ∃𝑥(𝜑 ∧ ∃𝑦𝜓))
 Colors of variables: wff setvar class Syntax hints:   ↔ wb 209   ∧ wa 399  ∃wex 1781 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 This theorem depends on definitions:  df-bi 210  df-an 400  df-ex 1782 This theorem is referenced by:  exdistrv  1956  19.42vv  1958  3exdistr  1962  sbccomlem  3778  coass  6099  uniuni  7488  eulerpartlemgvv  31866  bnj986  32459  dfiota3  33800  ac6s6f  35917
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