Theorem 19.42vv 1837

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 1836	. 2 |- ( E. x E. y ( ph /\ ps ) <-> E. x ( ph /\ E. y ps ) )
2		19.42v 1835	. 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 1427

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 1382  ax-gen 1384  ax-ie1 1428  ax-ie2 1429  ax-4 1446  ax-17 1465  ax-ial 1473

This theorem depends on definitions:  df-bi 116

This theorem is referenced by:  19.42vvv  1838  19.42vvvv  1839  exdistr2  1840  3exdistr  1841  ceqsex3v  2662  ceqsex4v  2663  elvvv  4514  dfoprab2  5710  resoprab  5755  ovi3  5795  ov6g  5796  oprabex3  5914  xpassen  6600  enq0enq  7044  enq0sym  7045  nqnq0pi  7051