Theorem ee4anv 1962

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 1687	. . 3 |- ( E. y E. z E. w ( ph /\ ps ) <-> E. z E. y E. w ( ph /\ ps ) )
2	1	exbii 1628	. 2 |- ( E. x E. y E. z E. w ( ph /\ ps ) <-> E. x E. z E. y E. w ( ph /\ ps ) )
3		eeanv 1960	. . 3 |- ( E. y E. w ( ph /\ ps ) <-> ( E. y ph /\ E. w ps ) )
4	3	2exbii 1629	. 2 |- ( E. x E. z E. y E. w ( ph /\ ps ) <-> E. x E. z ( E. y ph /\ E. w ps ) )
5		eeanv 1960	. 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 206	1 |- ( E. x E. y E. z E. w ( ph /\ ps ) <-> ( E. x E. y ph /\ E. z E. w ps ) )

Syntax hints:   /\ wa 104   <-> wb 105   E.wex 1515

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 1470  ax-7 1471  ax-gen 1472  ax-ie1 1516  ax-ie2 1517  ax-4 1533  ax-17 1549  ax-ial 1557

This theorem depends on definitions:  df-bi 117  df-nf 1484

This theorem is referenced by:  ee8anv  1963  cgsex4g  2809  th3qlem1  6724  dmaddpq  7492  dmmulpq  7493  ltdcnq  7510  enq0ref  7546  nqpnq0nq  7566  nqnq0a  7567  nqnq0m  7568  genpdisj  7636  axaddcl  7977  axmulcl  7979