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Theorem exdistr 1957
Description: Distribution of existential quantifiers. See also exdistrv 1958. (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 1956 . 2 (∃𝑦(𝜑𝜓) ↔ (𝜑 ∧ ∃𝑦𝜓))
21exbii 1849 1 (∃𝑥𝑦(𝜑𝜓) ↔ ∃𝑥(𝜑 ∧ ∃𝑦𝜓))
Colors of variables: wff setvar class
Syntax hints:  wb 205  wa 395  wex 1780
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1912
This theorem depends on definitions:  df-bi 206  df-an 396  df-ex 1781
This theorem is referenced by:  exdistrv  1958  19.42vv  1960  3exdistr  1963  rexcom4  3284  sbccomlem  3864  coass  6264  uniuni  7753  eulerpartlemgvv  33840  bnj986  34431  dfiota3  35366  ac6s6f  37507
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