Theorem 19.41vv 1832

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 1831	. . 3 |- ( E. y ( ph /\ ps ) <-> ( E. y ph /\ ps ) )
2	1	exbii 1542	. 2 |- ( E. x E. y ( ph /\ ps ) <-> E. x ( E. y ph /\ ps ) )
3		19.41v 1831	. 2 |- ( E. x ( E. y ph /\ ps ) <-> ( E. x E. y ph /\ ps ) )
4	2, 3	bitri 183	1 |- ( E. x E. y ( ph /\ ps ) <-> ( E. x E. y ph /\ 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:  19.41vvv  1833  rabxp  4487  rexiunxp  4591  mpt2mptx  5753  xpassen  6600  dmaddpqlem  6990  nqpi  6991  nqnq0pi  7051  nq0nn  7055