Theorem eean 1847

Description: Rearrange existential quantifiers. (Contributed by NM, 27-Oct-2010.) (Revised by Mario Carneiro, 6-Oct-2016.)

Hypotheses

Ref	Expression
eean.1	|- F/ y ph
eean.2	|- F/ x ps

Assertion

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

Proof of Theorem eean

Step	Hyp	Ref	Expression
1		eean.1	. . . 4 |- F/ y ph
2	1	19.42 1618	. . 3 |- ( E. y ( ph /\ ps ) <-> ( ph /\ E. y ps ) )
3	2	exbii 1536	. 2 |- ( E. x E. y ( ph /\ ps ) <-> E. x ( ph /\ E. y ps ) )
4		eean.2	. . . 4 |- F/ x ps
5	4	nfex 1568	. . 3 |- F/ x E. y ps
6	5	19.41 1616	. 2 |- ( E. x ( ph /\ E. y ps ) <-> ( E. x ph /\ E. y ps ) )
7	3, 6	bitri 182	1 |- ( E. x E. y ( ph /\ ps ) <-> ( E. x ph /\ E. y ps ) )

Syntax hints:   /\ wa 102   <-> wb 103   F/wnf 1389   E.wex 1421

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106  ax-5 1376  ax-7 1377  ax-gen 1378  ax-ie1 1422  ax-ie2 1423  ax-4 1440  ax-ial 1467

This theorem depends on definitions:  df-bi 115  df-nf 1390

This theorem is referenced by:  eeanv  1848  reean  2522