Theorem ee4anv 1927

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 1657	. . 3 |- ( E. y E. z E. w ( ph /\ ps ) <-> E. z E. y E. w ( ph /\ ps ) )
2	1	exbii 1598	. 2 |- ( E. x E. y E. z E. w ( ph /\ ps ) <-> E. x E. z E. y E. w ( ph /\ ps ) )
3		eeanv 1925	. . 3 |- ( E. y E. w ( ph /\ ps ) <-> ( E. y ph /\ E. w ps ) )
4	3	2exbii 1599	. 2 |- ( E. x E. z E. y E. w ( ph /\ ps ) <-> E. x E. z ( E. y ph /\ E. w ps ) )
5		eeanv 1925	. 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 1485

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 1440  ax-7 1441  ax-gen 1442  ax-ie1 1486  ax-ie2 1487  ax-4 1503  ax-17 1519  ax-ial 1527

This theorem depends on definitions:  df-bi 116  df-nf 1454

This theorem is referenced by:  ee8anv  1928  cgsex4g  2767  th3qlem1  6615  dmaddpq  7341  dmmulpq  7342  ltdcnq  7359  enq0ref  7395  nqpnq0nq  7415  nqnq0a  7416  nqnq0m  7417  genpdisj  7485  axaddcl  7826  axmulcl  7828