Theorem 19.42vv 1958

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 1956	. 2 |- ( E. x E. y ( ph /\ ps ) <-> E. x ( ph /\ E. y ps ) )
2		19.42v 1953	. 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 1538

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 1493  ax-gen 1495  ax-ie1 1539  ax-ie2 1540  ax-4 1556  ax-17 1572  ax-ial 1580

This theorem depends on definitions:  df-bi 117

This theorem is referenced by:  19.42vvv  1959  19.42vvvv  1960  exdistr2  1961  3exdistr  1962  ceqsex3v  2843  ceqsex4v  2844  elvvv  4782  dfoprab2  6051  resoprab  6100  ovi3  6142  ov6g  6143  oprabex3  6274  xpassen  6989  enq0enq  7618  enq0sym  7619  nqnq0pi  7625  axaddf  8055  axmulf  8056