Theorem 19.41vv 1915

Description: Theorem 19.41 of [Margaris] p. 90 with 2 quantifiers. (Contributed by NM, 30-Apr-1995.)

Assertion

Ref	Expression
19.41vv	|- ( E. x E. y ( ph /\ ps ) <-> ( E. x E. y ph /\ ps ) )

Distinct variable groups:   ps, x   ps, y

Allowed substitution hints:   ph( x, y)

Proof of Theorem 19.41vv

Step	Hyp	Ref	Expression
1		19.41v 1914	. . 3 |- ( E. y ( ph /\ ps ) <-> ( E. y ph /\ ps ) )
2	1	exbii 1616	. 2 |- ( E. x E. y ( ph /\ ps ) <-> E. x ( E. y ph /\ ps ) )
3		19.41v 1914	. 2 |- ( E. x ( E. y ph /\ ps ) <-> ( E. x E. y ph /\ ps ) )
4	2, 3	bitri 184	1 |- ( E. x E. y ( ph /\ ps ) <-> ( E. x E. y ph /\ ps ) )

Syntax hints:   /\ wa 104   <-> wb 105   E.wex 1503

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 1458  ax-gen 1460  ax-ie1 1504  ax-ie2 1505  ax-4 1521  ax-17 1537  ax-ial 1545

This theorem depends on definitions:  df-bi 117

This theorem is referenced by:  19.41vvv  1916  rabxp  4696  rexiunxp  4804  mpomptx  6009  xpassen  6884  dmaddpqlem  7437  nqpi  7438  nqnq0pi  7498  nq0nn  7502