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Theorem exdistrv 1882
Description: Distribute a pair of existential quantifiers (over disjoint variables) over a conjunction. Combination of 19.41v 1874 and 19.42v 1878. For a version with fewer disjoint variable conditions but requiring more axioms, see eeanv 1904. (Contributed by BJ, 30-Sep-2022.)
Assertion
Ref Expression
exdistrv  |-  ( E. x E. y (
ph  /\  ps )  <->  ( E. x ph  /\  E. y ps ) )
Distinct variable groups:    ph, y    ps, x    x, y
Allowed substitution hints:    ph( x)    ps( y)

Proof of Theorem exdistrv
StepHypRef Expression
1 exdistr 1881 . 2  |-  ( E. x E. y (
ph  /\  ps )  <->  E. x ( ph  /\  E. y ps ) )
2 19.41v 1874 . 2  |-  ( E. x ( ph  /\  E. y ps )  <->  ( E. x ph  /\  E. y ps ) )
31, 2bitri 183 1  |-  ( E. x E. y (
ph  /\  ps )  <->  ( E. x ph  /\  E. y ps ) )
Colors of variables: wff set class
Syntax hints:    /\ wa 103    <-> wb 104   E.wex 1468
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-5 1423  ax-gen 1425  ax-ie1 1469  ax-ie2 1470  ax-4 1487  ax-17 1506  ax-ial 1514
This theorem depends on definitions:  df-bi 116
This theorem is referenced by:  prodmodc  11359  txbasval  12450
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