Theorem exdistr 1784

Description: Distribution of existential quantifiers. (Contributed by NM, 9-Mar-1995.)

Assertion

Ref	Expression
exdistr	|- (E.xE.y(ph /\ ps) <-> E.x(ph /\ E.yps))

Distinct variable group:   ph,y

Allowed substitution hints:   ph(x)   ps(x,y)

Proof of Theorem exdistr

Step	Hyp	Ref	Expression
1		19.42v 1783	. 2 |- (E.y(ph /\ ps) <-> (ph /\ E.yps))
2	1	exbii 1493	1 |- (E.xE.y(ph /\ ps) <-> E.x(ph /\ E.yps))

Syntax hints:   /\ wa 97   <-> wb 98  E.wex 1378

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 1333  ax-gen 1335  ax-ie1 1379  ax-ie2 1380  ax-4 1397  ax-17 1416  ax-ial 1424

This theorem depends on definitions:  df-bi 110

This theorem is referenced by:  19.42vv  1785  3exdistr  1789  sbel2x  1871  sbexyz  1876  sbccomlem  2826  uniuni  4149  coass  4782  subhalfnqq  6397