Theorem ee4anv 1922

Description: Rearrange existential quantifiers. (Contributed by NM, 31-Jul-1995.)

Assertion

Ref	Expression
ee4anv	|- ( E. x E. y E. z E. w ( ph /\ ps ) <-> ( E. x E. y ph /\ E. z E. w ps ) )

Distinct variable groups:   ph, z   ph, w   ps, x   ps, y   y, z   x, w

Allowed substitution hints:   ph( x, y)   ps( z, w)

Proof of Theorem ee4anv

Step	Hyp	Ref	Expression
1		excom 1652	. . 3 |- ( E. y E. z E. w ( ph /\ ps ) <-> E. z E. y E. w ( ph /\ ps ) )
2	1	exbii 1593	. 2 |- ( E. x E. y E. z E. w ( ph /\ ps ) <-> E. x E. z E. y E. w ( ph /\ ps ) )
3		eeanv 1920	. . 3 |- ( E. y E. w ( ph /\ ps ) <-> ( E. y ph /\ E. w ps ) )
4	3	2exbii 1594	. 2 |- ( E. x E. z E. y E. w ( ph /\ ps ) <-> E. x E. z ( E. y ph /\ E. w ps ) )
5		eeanv 1920	. 2 |- ( E. x E. z ( E. y ph /\ E. w ps ) <-> ( E. x E. y ph /\ E. z E. w ps ) )
6	2, 4, 5	3bitri 205	1 |- ( E. x E. y E. z E. w ( ph /\ ps ) <-> ( E. x E. y ph /\ E. z E. w ps ) )

Syntax hints:   /\ wa 103   <-> wb 104   E.wex 1480

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-5 1435  ax-7 1436  ax-gen 1437  ax-ie1 1481  ax-ie2 1482  ax-4 1498  ax-17 1514  ax-ial 1522

This theorem depends on definitions:  df-bi 116  df-nf 1449

This theorem is referenced by:  ee8anv  1923  cgsex4g  2763  th3qlem1  6603  dmaddpq  7320  dmmulpq  7321  ltdcnq  7338  enq0ref  7374  nqpnq0nq  7394  nqnq0a  7395  nqnq0m  7396  genpdisj  7464  axaddcl  7805  axmulcl  7807