Theorem exdistr 1861

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

Step	Hyp	Ref	Expression
1		19.42v 1860	. 2 |- ( E. y ( ph /\ ps ) <-> ( ph /\ E. y ps ) )
2	1	exbii 1567	1 |- ( E. x E. y ( ph /\ ps ) <-> E. x ( ph /\ E. y ps ) )

Syntax hints:   /\ wa 103   <-> wb 104   E.wex 1451

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-5 1406  ax-gen 1408  ax-ie1 1452  ax-ie2 1453  ax-4 1470  ax-17 1489  ax-ial 1497

This theorem depends on definitions:  df-bi 116

This theorem is referenced by:  exdistrv  1862  19.42vv  1863  3exdistr  1867  sbel2x  1949  sbexyz  1954  sbccomlem  2951  uniuni  4332  coass  5015  subhalfnqq  7170