Theorem 19.42vv 1963

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 1961	. 2 |- ( E. x E. y ( ph /\ ps ) <-> E. x ( ph /\ E. y ps ) )
2		19.42v 1958	. 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 1541

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 1496  ax-gen 1498  ax-ie1 1542  ax-ie2 1543  ax-4 1559  ax-17 1575  ax-ial 1583

This theorem depends on definitions:  df-bi 117

This theorem is referenced by:  19.42vvv  1964  19.42vvvv  1965  exdistr2  1966  3exdistr  1967  ceqsex3v  2859  ceqsex4v  2860  elvvv  4818  dfoprab2  6108  resoprab  6157  ovi3  6199  ov6g  6200  oprabex3  6335  xpassen  7094  enq0enq  7762  enq0sym  7763  nqnq0pi  7769  axaddf  8199  axmulf  8200