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Theorem exdistr 1787
 Description: Distribution of existential quantifiers. (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 1786 . 2 (∃𝑦(𝜑𝜓) ↔ (𝜑 ∧ ∃𝑦𝜓))
21exbii 1496 1 (∃𝑥𝑦(𝜑𝜓) ↔ ∃𝑥(𝜑 ∧ ∃𝑦𝜓))
 Colors of variables: wff set class Syntax hints:   ∧ wa 97   ↔ wb 98  ∃wex 1381 This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 99  ax-ia2 100  ax-ia3 101  ax-5 1336  ax-gen 1338  ax-ie1 1382  ax-ie2 1383  ax-4 1400  ax-17 1419  ax-ial 1427 This theorem depends on definitions:  df-bi 110 This theorem is referenced by:  19.42vv  1788  3exdistr  1792  sbel2x  1874  sbexyz  1879  sbccomlem  2832  uniuni  4183  coass  4839  subhalfnqq  6512
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