Metamath Proof Explorer < Previous   Next > Nearby theorems Mirrors  >  Home  >  MPE Home  >  Th. List  >  exdistr Structured version   Visualization version   GIF version

Theorem exdistr 1916
 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 1915 . 2 (∃𝑦(𝜑𝜓) ↔ (𝜑 ∧ ∃𝑦𝜓))
21exbii 1771 1 (∃𝑥𝑦(𝜑𝜓) ↔ ∃𝑥(𝜑 ∧ ∃𝑦𝜓))
 Colors of variables: wff setvar class Syntax hints:   ↔ wb 196   ∧ wa 384  ∃wex 1701 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1719  ax-4 1734  ax-5 1836  ax-6 1885 This theorem depends on definitions:  df-bi 197  df-an 386  df-ex 1702 This theorem is referenced by:  19.42vv  1917  3exdistr  1920  sbccomlem  3490  coass  5613  uniuni  6920  eulerpartlemgvv  30219  bnj986  30732  dfiota3  31672  ac6s6f  33613
 Copyright terms: Public domain W3C validator