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

Theorem exdistr 1950
Description: Distribution of existential quantifiers. (Contributed by NM, 9-Mar-1995.)
Assertion
Ref Expression
exdistr  |-  ( E. x E. y (
ph  /\  ps )  <->  E. x ( ph  /\  E. y ps ) )
Distinct variable group:    ph, y
Allowed substitution hints:    ph( x)    ps( x, y)

Proof of Theorem exdistr
StepHypRef Expression
1 19.42v 1949 . 2  |-  ( E. y ( ph  /\  ps )  <->  ( ph  /\  E. y ps ) )
21exbii 1644 1  |-  ( E. x E. y (
ph  /\  ps )  <->  E. x ( ph  /\  E. y ps ) )
Colors of variables: wff setvar class
Syntax hints:    <-> wb 184    /\ wa 369   E.wex 1596
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1601  ax-4 1612  ax-5 1680  ax-6 1719  ax-7 1739  ax-10 1786  ax-12 1803
This theorem depends on definitions:  df-bi 185  df-an 371  df-ex 1597  df-nf 1600
This theorem is referenced by:  19.42vv  1951  3exdistr  1954  sbel2xOLD  2192  sbccomlem  3410  coass  5524  uniuni  6587  eulerpartlemgvv  27955  dfiota3  29150  ac6s6f  30185  bnj986  33091
  Copyright terms: Public domain W3C validator