Theorem 19.42vv 1904

Description: Theorem 19.42 of [Margaris] p. 90 with 2 quantifiers. (Contributed by NM, 16-Mar-1995.)

Assertion

Ref	Expression
19.42vv	|- ( E. x E. y ( ph /\ ps ) <-> ( ph /\ E. x E. y ps ) )

Distinct variable groups:   ph, x   ph, y

Allowed substitution hints:   ps( x, y)

Proof of Theorem 19.42vv

Step	Hyp	Ref	Expression
1		exdistr 1902	. 2 |- ( E. x E. y ( ph /\ ps ) <-> E. x ( ph /\ E. y ps ) )
2		19.42v 1899	. 2 |- ( E. x ( ph /\ E. y ps ) <-> ( ph /\ E. x E. y ps ) )
3	1, 2	bitri 183	1 |- ( E. x E. y ( ph /\ ps ) <-> ( ph /\ E. x E. y ps ) )

Syntax hints:   /\ wa 103   <-> wb 104   E.wex 1485

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 1440  ax-gen 1442  ax-ie1 1486  ax-ie2 1487  ax-4 1503  ax-17 1519  ax-ial 1527

This theorem depends on definitions:  df-bi 116

This theorem is referenced by:  19.42vvv  1905  19.42vvvv  1906  exdistr2  1907  3exdistr  1908  ceqsex3v  2772  ceqsex4v  2773  elvvv  4672  dfoprab2  5897  resoprab  5946  ovi3  5986  ov6g  5987  oprabex3  6105  xpassen  6804  enq0enq  7380  enq0sym  7381  nqnq0pi  7387  axaddf  7817  axmulf  7818