Theorem 19.42vv 1922

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 1920	. 2 |- ( E. x E. y ( ph /\ ps ) <-> E. x ( ph /\ E. y ps ) )
2		19.42v 1917	. 2 |- ( E. x ( ph /\ E. y ps ) <-> ( ph /\ E. x E. y ps ) )
3	1, 2	bitri 184	1 |- ( E. x E. y ( ph /\ ps ) <-> ( ph /\ E. x E. y ps ) )

Syntax hints:   /\ wa 104   <-> wb 105   E.wex 1502

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-5 1457  ax-gen 1459  ax-ie1 1503  ax-ie2 1504  ax-4 1520  ax-17 1536  ax-ial 1544

This theorem depends on definitions:  df-bi 117

This theorem is referenced by:  19.42vvv  1923  19.42vvvv  1924  exdistr2  1925  3exdistr  1926  ceqsex3v  2793  ceqsex4v  2794  elvvv  4703  dfoprab2  5937  resoprab  5986  ovi3  6027  ov6g  6028  oprabex3  6147  xpassen  6847  enq0enq  7447  enq0sym  7448  nqnq0pi  7454  axaddf  7884  axmulf  7885