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Theorem exdistr 1950
Description: Distribution of existential quantifiers. See also exdistrv 1951. (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 1949 . 2 (∃𝑦(𝜑𝜓) ↔ (𝜑 ∧ ∃𝑦𝜓))
21exbii 1842 1 (∃𝑥𝑦(𝜑𝜓) ↔ ∃𝑥(𝜑 ∧ ∃𝑦𝜓))
Colors of variables: wff setvar class
Syntax hints:  wb 205  wa 395  wex 1773
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1905
This theorem depends on definitions:  df-bi 206  df-an 396  df-ex 1774
This theorem is referenced by:  exdistrv  1951  19.42vv  1953  3exdistr  1956  rexcom4  3279  sbccomlem  3859  coass  6257  uniuni  7745  eulerpartlemgvv  33904  bnj986  34494  dfiota3  35427  ac6s6f  37553
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