Theorem eean 1959

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 1711	. . 3 |- ( E. y ( ph /\ ps ) <-> ( ph /\ E. y ps ) )
3	2	exbii 1628	. 2 |- ( E. x E. y ( ph /\ ps ) <-> E. x ( ph /\ E. y ps ) )
4		eean.2	. . . 4 |- F/ x ps
5	4	nfex 1660	. . 3 |- F/ x E. y ps
6	5	19.41 1709	. 2 |- ( E. x ( ph /\ E. y ps ) <-> ( E. x ph /\ E. y ps ) )
7	3, 6	bitri 184	1 |- ( E. x E. y ( ph /\ ps ) <-> ( E. x ph /\ E. y ps ) )

Syntax hints:   /\ wa 104   <-> wb 105   F/wnf 1483   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-ial 1557

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

This theorem is referenced by:  eeanv  1960  reean  2675